Is my proof of the uniqueness of $0$ non-circular?

The "proof" of the lemma is incorrect: when you write "let $x=y$," you are assuming what you are trying to prove. You need to start with "$x+z=y+z$" and deduce "$x=y$."

As a matter of fact, I'm worried the axioms you've given are not enough, although I don't immediately see how to construct a model proving this. Apostol does give other axioms (the order axioms), but they seem to take the uniqueness of $0$ for granted. I wouldn't be too surprised if there is a clever way to show that $0_x=0_y$ just from the axioms he gives, but I can't see it right now, and I also wouldn't be surprised if it's his mistaske.

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Note that the existence of a specific zero element is usually taken as one of the field axioms; see https://en.wikipedia.org/wiki/Field_(mathematics)#Definition_and_illustration.