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I am trying to find a clear definition of a formal fallacy and an informal fallacy.
Wikipedia gives the following definition:
Informal fallacies – arguments that are fallacious for reasons other than structural (formal) flaws and usually require examination of the argument's content
This definition is not particularly clear to me. Wikipedia provides a list of fallacies, but why isn't Appeal to Ignorance or Begging the Question a "structural" flaw?
The wikipedia definition you quote strikes me as being not particularly helpful for grasping the distinction (though not particularly wrong).
A better way to put it is that formal fallacies means fallacious inferences in formal logic. Here, we'd put things like affirming the consequent:
For a formal fallacy, no rule of inference enables the leap from these premises to the conclusion and the truth table shows that it's possible to have the opposite conclusion while both premises are still true.
Maybe to reword that, under normal sentential logic, any argument that is invalid (= able to have a false conclusion with all true premises) is committing a formal fallacy whether or not that fallacy has a name.
Conversely, any other error in reasoning can be called "fallacious" but this would be an informal fallacy. You're right that these can also be about the structure of the argument.
Thinking about it a bit, several things that are informal fallacies also contain a formal fallacy as well. For instance, argument from ignorance when symbolized would be formally fallacious. But argument from ignorance is not a formal fallacy. The explanation for this might sound like a sleight of hand, but you can only commit a formal fallacy when you make a formal (i.e. deductive, modal, truth-functional, propositional and their ilk) argument.
Conversely, you can commit an informal fallacy in any sort of argumentation. Thus, slippery slope formalizes to the chain rule / a series of hypothetical syllogisms, but we still consider it an informal fallacy. Presumably, this is because we take the series of hypotheticals in question to have dubious veracity.
A second feature that happens because informal fallacies apply to informal argument is that there can be a large amount of disagreement as to whether or not a given informal fallacy has been committed. (In contrast, if we receive the already symbolized deductive argument, there's no question as to whether or not it yields its conclusion in a truth-preserving fashion). Thus, we can argue endlessly about whether something is "ad hominem" or relevant to the case in question.
Or as we've seen several times on this site, there can be disagreement about whether a particular claim is "question-begging." I recall a question that contained an argument roughly of this form:
The OP didn't see how this was question-begging but myself and several others pointed out that the premise 1 captures the entire argument. If we formalize the argument, it turns out to be valid, but that does not mean the argument as supplied isn't question begging).
formal fallacy (1) applies to formal argument (2) is objectively clear
informal fallacy (1) applies to any type of argument (2) requires a judgment as to whether it is fallacious (3) can produce an argument that would be valid if symbolized (or can fail to)
The whole idea of informal logic is flawed.
A conclusion in an informal argument does not logically follow from the premises, by definition. If you believe the argument anyway, it's because you are, perhaps unconsciously, adding some additional assumptions or in general you believe it for other unrelated reasons.
There is a lot of ambiguity in natural language which allows for a range of interpretations so what seems 'logical' to one person might not seem so to another.
