How to properly claim credit for peer review?

The standard information to give is the journal's name and only this. As far as I recall I have never seen a CV that gives more details than that (mostly I know academic CVs of mathematicians).

I would not go as far as saying that it would be unethical to include more information, but it would seem unusual and the advantage is not very clear.

If you are concerned about the claim not being credible, I'd not worry too much about this. However, some journals and publishers offer certificates for reviewers. See for example this information about reviewer recognition at Elsevier

There may be exceptions to this if you happen to be involved in some very high profile reviewing endeavor, but if you reviewed a normal paper I'd just give the journal's name.

I would recommend against doing this unless the journal editor or conference chair tells you it is ok. I assume the review was blind or the question wouldn't arise. But blind reviews are intended to remain blind.

But I wonder what your motivation is here. Why not just state in your CV that you have been a reviewer for [journal name] or [conference name year]? Reviewing is intended to be a service to the community, not a way to associate yourself with someone else's work.

My own reaction to seeing a specific claim on a CV would not be favorable, I think.

It may be, in some cases that an author will want to thank the reviewers for helpful comments, but even then, it is very rare that names are used, or even known.

If you want to change the reviewing process, on the other hand, removing the "blind" aspect of it, I'd suggest that you do it directly and openly, not by just doing it. There are venues with open reviews.

But ask the editor.

I have recently started using this service called Publons (https://publons.com/about/home/) by Clarivate Analytics. They request that you forward peer-review acknowledgements to a certain email address, and they show up on your personal profile page after verification with sensitive details anonymized. It might help to just link to your profile on your CV or use the list they provide.