I had to deliver exactly that some months ago. It is right what you described, and it makes not much sense at all, but that's what they want.
A 'notary public' is a very common side-qualification, every office has typically one or two people that carry it. What you do is
- make photocopies of your (Green Card),
- the go to a Notary Public
- show your ID, and sign in front of the notary public that you confirm that the copy is identical with the original
- they give you a seal that confirms that they saw your ID and they saw you signing.
Note that the seal does not need to be on the photocopy (but it can).
This was accepted without questions, and - I asked them - that's what they wanted.
Of course, that leaves various obvious avenues of cheating - you can bring a picture of your dog instead of the requested photocopy, and the notary public would not care; and you can replace the photocopies after the sealed stamp is signed, as they are on different - not physically connected - sheets of paper.
The core point is that you signed it, and the notary just verifies that is was you that signed. If anything turns out wrong or fishy, they know whom to talk to - you signed it for correctness.
There are probably better ways - like have the notary public make the photocopies and put the seal physically onto them - but that is not what is done in the USA. For whatever reasons.