Just to be nitpicky, 3 SAT is NP-Complete which means its NP-hard.

The NP-hard property allows you to reduce any problem in NP to it.

As you stated it, generalizing (or reducing) 3-SAT to another problem proves nothing and is a common mistake for undergraduates learning complexity theory (reducing the wrong way).

He said generalize it (the algorithm) to other NP problems, which I took to mean 'turn every NP problem into 3-SAT and solve that'

The word generalize is meaningless and undefined, which is what led to your confusion as well as multiple interpretations (which is one of the things I was picking at).

To say you generalize an NP problem isn't even a statement that typechecks in the language of complexity.

Reduction on the other hand, is a formally defined term and is very specific in what it means exactly.

I agree you can't generalize a problem, but if you make an algorithm work on more than just one specific problem class, aren't you generalizing the algorithm (in this case, by reducing the problem)? The only other word I can think to use there instead of generalize would be 'generify' but that's only used in the context of languages that have generics, IIRC.

If moultano had actually said 'generalize [the problem]' then I wouldn't have even left a comment, because everything you have said is correct, I just don't think his statement needed that clarification. Now, however, I'm interested in whether the word 'generalize' can be used with respect to algorithms. (<--- someone not in complexity theory)