I am extremely skeptical of the idea that this will help further useful debate online. I am actually skeptical of the usefulness of the "fallacy" frame in general. I mainly find that people who accuse other people of committing a particular fallacy are just using their "intro to philosophy" chops to bully someone who actually has a decent point.
Even when someone does commit a fallacy, pointing out the specific fallacy they made is counterproductive. Fallacies are inherently dismissive; they blow off the other person in the argument. If the other person isn't trained in the terms of logical fallacies, they're left with no way to continue their argument, because they don't really even understand the criticism.
It is almost always better, in my opinion, to rely on an actual explanation than to rely on a fallacy catchphrase. If you find yourself wanting to call a fallacy on someone, instead strip out the language of fallacy, and focus on the specific example of why they are wrong in the given case. Don't accuse someone of committing a strawman fallacy; instead, just point out how they overlooked an important part of your position. Don't accuse someone of committing an ad hominem fallacy; just ask them why they think your personal flaws have compromised your opinion on the matter at hand.
The core of honest and productive argument is engagement. Both sides have to make an effort to understand the other, and both sides have to show a certain basic respect for the other side as a person. The language of logical fallacy is highly disengaging. It is dismissive, and it shows a lack of respect for the other person.
Also, a lot of so-called fallacies are really not fallacies at all, or at least deserve more of your time than a simple dismissal. For further explanation, I'm going to go through the list of fallacies from the website above and look at each one and see whether it's really worth keeping around.
1) Strawman fallacy
People accuse others of committing this fallacy all the time. In my experience though, deliberate strawman arguments are very rare. Usually, the strawman fallacy comes up because of a genuine misunderstanding. The proper solution is not to attack the opponent for fallacious reasoning, but rather to restate your real position, and point out how it deals with whatever problems they pointed out.
2) Gambler's fallacy
This is a genuine failure of ordinary human reasoning. I hardly ever see it come up in an argument though; usually it just sits underneath normal conscious thought and influences decisions. If someone actually leans on this fallacy to make an argument, then they don't understand how probability works. Probabilistic reasoning is its own complex worldview, so the sad truth is you probably won't be able to convince someone using this fallacy that they're wrong without making them sit through an entire course on basic probability.
3) False cause
People use this a lot, and I do think it is fair to point out, "correlation does not imply causation." However, it is even better to counter the opposing argument by proposing a different theory of the causal relationships involved in the case, and showing that your theory explains the correlation just as well if not better than the opposing theory.
Correlation is cause enough to call for an explanatory theory. If your opponent has developed such a theory, you need to have one too. Otherwise, you should concede that your opponent has a point.
4) Appeal to emotion
This is not a fallacy at all. In fact, every argument necessarily involves some level of appeal to emotion. David Hume points out to us that purely logical reasoning cannot actually motivate anything. In his words, "It is not contrary to reason to prefer the destruction of the whole world to the scratching of my finger."
In order to support an argument that it is worth scratching his finger to save the world, it is necessary to introduce the premise that he should in fact care about the world. Any such premise would have to be an appeal to emotion. An argument with no appeal to emotion has no power to motivate action, since only emotions can motivate action.
5) The fallacy fallacy
Also known as fallacy inception. If someone has gone so far up their own ass that they're actually arguing this way, just walk away. Nothing good will come of it.
6) Slippery slope
This is another fallacy that I feel is more of a misunderstanding than a true fallacy. The actual flaw of a slippery slope argument is that there is no causal mechanism given for how the first step will lead to further movement down the slope. Usually, the causal mechanism is omitted because the person making the argument feels that it is obvious and doesn't bear mentioning.
The proper response is to ask for a deeper explanation of the causal mechanism that will lead to the slipperiness of the slope. Then criticize the causal mechanism they provide, or accept it and cede the point.
7) Ad hominem
Sometimes, ad hominem arguments have a point. If someone is trying to convince you to invest in a particular stock, it is fair to point out to them that their previous five stock tips all proved disastrous. In general, if your argument relies on a claim that you have special information, experience, or insight, then an ad hominem attack is a valid criticism.
If someone does attack you ad hominem, the proper counter is to show that your argument can stand without relying on any special insight or special information from you. If they nevertheless continue with ad hominem argumentation, then it is fair to call them out for disingenuous personal attacks.
8) Tu quoque
Although this move can be diversionary, it's not really a fallacy. More just an annoying debate tactic. The proper response is to address the issue raised, then move back to the original argument.
If your opponent continues to bring up diversions in a debate, then they're arguing in bad faith and the best response is to walk away. There is nothing to be gained from trying to engage with someone who is deliberately disengaging you. If there is an audience to such an exchange, then point out the diversionary nature of the opponent's tactics.
9) Personal incredulity
If someone doesn't get your argument, then your job is to explain it better. If you don't think you can explain it better, then it's time to walk away. There's nothing to be gained from debating someone who is literally not capable of understanding your argument. If they're being incredulous as a debate tactic, then walking away is even more appropriate.
Consider, also, that there may be a worldview conflict. It might be that you and your opponent are approaching the question from very different worldviews. Debating across worldviews can be useful sometimes, although I think that worldview conflicts are better handled through art and literature than through logical argument.
10) Special pleading
Accusing someone of this is actually a form of ad hominem attack. It is saying that their new, modified position must be incorrect because they changed from their old position. Focus on the position itself in your argument, not how they came to that position. If they keep changing positions, then tell your opponent to clearly state their position, and make them clearly state whatever concessions or modifications they're making in response to your argument.
Note also: one of the points of argumentation is to polish and clarify positions. If nobody ever modifies their position based on an argument, then what was the point?
11) Loaded question
I'm not certain what would constitute a "loaded question." Every question has to make certain assumptions in order to make any sense. If someone asks you a question that assumes something you don't accept, then point out the assumption and explain why you don't accept it. Getting down to assumptions is part of why we debate.
12) Burden of proof
If someone believes something and you want to change that belief, the burden of proof is on you to show them why they should change. If someone is trying to change your belief, the burden of proof is on them. There's really not much more to it. I'm honestly not sure where the so-called "fallacy" is here.
13) Ambiguity
This is a real fallacy in argumentation. However, most people who commit it don't realize what they're doing. The proper response is to point it out and explain the appropriate distinction. If the opponent does this on purpose, then either they're making a funny joke or they're not worth debating.
14) Bandwagon
This isn't a fallacy at all. In general, if a lot of people believe something, there's probably a good reason for it. Bandwagon reasoning doesn't clinch an argument by itself, but it certainly doesn't hurt. If your opponent makes a bandwagon argument, then you absolutely have to address how it came to be that so many people are wrong.
15) Appeal to authority
This also isn't a fallacy. Certain people are respected by society because they are known to have solid, well-thought positions, especially in their areas of expertise. You cannot simply dismiss that fact. If your position is contradicted by a prominent and respected authority, it is your responsibility to show why the authority is wrong.
16) Fallacy of composition/division
This is a true fallacy. Like most other real fallacies, chances are your opponent does not realize they have committed a fallacy. The proper response is to clearly explain the proper distinction, and show why the property of the part does not have to apply to the whole, or why the property of the whole does not have to apply to the part.
17) No true scotsman
The proper response here is to force your opponent to define their terms. In the classic scotsman example, you should make the opponent define scotsman. One of two things will happen: they will either define scotsman so that their position is necessarily true, or they will present a definition to which you can provide a counter example.
18) Genetic fallacy
This is very similar to ad hominem. The right response is to divorce the argument from its origin by clearly showing how all of the information necessary to the argument is well-supported outside the context of the argument's shady origins. Keep in mind though, if your argument depends on information from a shady source, then your opponent has a point in questioning that source.
19) Black or white
The only good response to a black or white argument is to clearly lay out an alternative. If you can't provide a reasonable alternative, then your opponent has a point.
20) Begging the question
The nature of a logical argument is that the conclusion is contained in the premises. Every logical argument assumes its conclusion. The purpose of a logical argument is to demonstrate the connections between ideas. Logical arguments move us because they show how assumptions we already hold require us to also hold the conclusion of the argument.
Begging the question is essentially when the connections pointed out are trivial. If the premise is too similar to the conclusion, and we don't already believe in the conclusion, then there's no compelling reason to accept that the premise is true. Begging the question is, however, a subjective evaluation, not an objective evaluation. Whether or not a connection is trivial is in the eye of the beholder. When presented with a logical argument that seems to be begging the question, explain why the connections presented in the argument are trivial and unmoving to you.
21) Appeal to nature
The real point of an appeal to nature is that we have a good understanding of the consequences of longstanding "natural" approaches, while we often have a poorer understanding of the consequences of newer "artificial" approaches. It is up to you to show that there is in fact an adequate understanding of the "artificial" approach, and that the benefits outweigh the costs and risks. If you can't do that, then your opponent has a point.
22) Anecdotal fallacy
If someone uses this line of argumentation, it's your responsibility to demonstrate why their case should not be taken as representative. If you can't do that, then your opponent has a point.
23) The Texas sharpshooter
This is very similar to the false cause fallacy, and the proper response is similar. If your opponent has an explanation for a correlation, then you need to have a counter explanation ready. If you really think the correlation is just coincidental, then you need to use statistical methods or additional data-gathering to prove your point.
24) Middle ground fallacy
This is one of the most difficult fallacies to beat in an argument. The best you can do is to repeat and refine your reasoning for why a position on the extreme is in fact the better position. This is one of the few cases where I would advocate explicitly pointing out the fallacy your opponent is committing. That is because many people view a tendency toward the middle to be a virtue, not a fallacy at all, and it is difficult to fight that preconception without reframing the problem.
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