I got a request to help comment on a paper in a journal from a professor. It's an informal request -- the professor is the one who officially reviews the paper. Can I put "reviewer" in my CV or not?
I think not. Ask yourself what the editor of the journal would think if your CV crossed their desk, and they saw you listing yourself as a reviewer and they'd never heard your name before. Having a review delegated to you does not make you a reviewer for that journal. The journal chose your professor, not you, so you should not muddy the waters about what you've been asked to do. If someone asks you about your reviewing experience, feel free to tell them that you did some subreviewing for your professor.
For the purpose you describe, the answer is no.
However, I would say that in reference to the headline question; yes, you are 'a reviewer'. You are simply not a reviewer in the traditional (but narrow) sense of somebody invited by an editor to carry out pre-publication peer review.
Anybody can be a reviewer. Carry out some post-publication peer-review and upload it to an appropriate website (for example, Publons). Then you will have a verifiable record as a peer reviewer. You should not upload the pre-publication review you mention as I expect it was shared with you in strict confidence.