What Fallacy Is This? Focusing On An Insignificant Detail

I was going to ask this question, but this is probably the selective attention or the argument by selective reading which I don't know how to differentiate, it one makes argument x y z but only x is falsified and therefore they are wrong which is not inherently true. A red herring is still a correct term but I think it is more vague.

Yes, I think River may be right.  Selective Argument and Argument by Selective Reading sound like it.

These 2 sound very similar; what's the difference?

selective attention doesn't necessarily imply focusing on insignificant details. Just that part of an argument is responded to, and the rest isn't.

argument by selective reading also doesn't imply nitpicking (though it does include it); just that, if someone made several claims or arguments, the weakest one is taken as if it were the best one, so when it is refuted, you act as if all the claims/arguments were refuted.

You're right.  The Nutpicking Fallacy is something completely different.

I also saw discussion of Nitpicking--focusing on the inaccuracy of a specific detail in someone's argument and making a big deal out of it.   That sounds like a Red Herring fallacy because the intent of focusing on the detail is to distract from the issue being discussed  (example: the U.S. has wasted $2 billion on some project.   There should be more accountability.  Response:  "actually, it was $1.85 billion. You should get your facts straight.")

That's basically what nitpicking is.   My questions was slightly different;  the responder doesn't nitpik about the accuracy of some detail, such as a number, but rather focuses on a relatively insignificant part of the other's argument.   

Example:  We should support the proposed budget which will provide important services for our community.    Response:   It's a terrible proposal.  It doesn't do anything to address...(some relatively insignificant issue which the responder harps on at length).