https://www.lingvolive.com/ru-ru/translate/ru-en/удостоверение%20личности

Сертификат удостоверяет благодарность от руководства (почётная грамота), пройденное обучение, квалификацию, но чтобы личность... 😳

ID is correct because it stands for IDENTIFICATION, which correctly translates the original phrase. But PROOF OF ID is not correct because the correct phrase should be PROOF OF IDENTITY, and ID doesn’t stand for IDENTITY. In other words IDENTIFICATION = PROOF OF IDENTITY.

And yes, you will find “proof of ID” and even “proof of identification” online, but they’re erroneous - written by people who don’t know the difference.

удалил
Wiki even says ID = Identity document

Good to know. Yeah, the card itself is called IDENTIFICATION and it contains PROOF OF IDENTITY. I thought this was going to be confusing to you, but I’m glad you got it 👍🏼