I think you can go through any of the books you listed and get an understanding of proofs. The proofs books usually use basic number theory or set theory for examples because the point is to learn the logic and structure of proofs. (If the book isn't enough for you, you can use Khan Academy as a refresher.) Learning more math is left to other books.
I looked through the table of contents for the ones you've linked, but neither looks like an upper year math book. Typically, if there is an explicit course on proofs, it's covered in the lower division.
The proofs course I took used "Mathematical Proofs: A Transition to Advanced Mathematics" by Chartrand, Polimeni, and Zhang. It's ridiculously expensive on Amazon right now, so I'm not sure I can recommend it for the cost. But it does cover more areas of math using proofs. It covers proofs in set theory, number theory, combinatorics, calculus, and group theory at a very basic level. The books you linked seem to cover less of those topics, so I think Chartrand et al. is a more gentle intro to different branches of upper division math.