Kaiden writes: I don't know what "truth claim" would refer to, other than the conclusion of the argument. It is my best guess, since I do not have prior experience with the phrase "truth claim of an argument". I do not necessarily agree with the way the definition is formulated (again, I don't even use the term "truth claim"). To acknowledge your objection, though, I copied and pasted your definition into Google and found that Craig Wright and Dr. Bennett both use the exact definition for the Genetic Fallacy that your Question presents. Dr. Bennett might provide a deeper explanation for us here of his definition. I did go to Dr. Bennett's archive entry to see how he lays out the logical form of this fallacy. Based on it, I might be able to address your objection to an extent, on behalf of Wright and Dr. Bennett.
Dr. Bennett puts the logic form of the Genetic Fallacy in this way:
The origin of the claim is presented.
Therefore, the claim is true/false.
Notice the word "therefore". This word indicates that the above passage is an argument. Dr. Bennett is teaching that the fallacy is a mistake in reasoning . That is to say, the fallacy pertains to making an argument or an inference, rather than to evaluating an argument. Due to this, the two suggestions that you give us at the end of your comment should not be taken up. For those suggestions would have us erroneously regard the fallacy as a mistake that pertains to evaluating arguments. Of course, the fallacy might occur in an evaluation, but only insofar as the evaluation involves making an argument.
You said, "I have an objection with the 'Basing the conclusion of an argument on the origin of its claims or premises' though, because the conclusion, since it is a part of an argument, is based on the premises, and the fallacy is that we evaluate the source rather than the claims or premises."
Dr. Bennett might give a better explanation of what he means by "truth claim", and if it does refer to the conclusion of the argument, then he might offer a better answer to your objection than I can. It is not a very clearly articulated definition, and it is unsurprising that we are struggling to make sense of it. But here's what I'll say. If "truth claim" refers to the conclusion, I think there is a way to clarify and alter the wording of the definition in response to your objection. Notice again that the form of the fallacy is an argument form. The conclusion says that the claim in question is true/false. This conclusion is based on the premise of the argument, as you correctly note. But the point of the fallacy has to do with what the premise states. The premise states the claim's origin, and it is from this statement alone that the conclusion is inferred. Thus, the Genetic fallacy is "basing the conclusion of an argument (a conclusion that says that a given claim is true/false) on the origin of the claim (on a premise that only states the claim's origin)." This might be a way of understanding Dr. Bennett and Wright's definition.
Thank you for your thoughtful objection, Kostas.