Ugh. My one encounter with an American notary involved a bizarre statement on his part.
My university insisted that I got Document X notarised (sorry, I don't remember what it was), so off I went in search of a notary. I found one near where I lived, and went to see him.
He went through what needed to be done, and then said that I had to sign the paper in front of him. OK, fair enough, that way he knows that I signed it. So I signed it in the real-world equivalent of "S T Cynic", which he didn't like. So far, so ... um ... whatever.
Then he complained that "S T Cynic" was "not a signature". He didn't say, "Sorry, I meant that you have to sign your full name," or something like that. No, he decided that something that the INS (for my pink laminated plastic thing(1)), the British passport authorities, all my banks, my credit cards, American Express(2), and in fact every "authority" situation I had encountered since I was about seven years old ALL accepted as my signature was, in fact, NOT a signature.
(1) Better known as a "green card" although at the time I got mine, 1984 or so, none of it was green and it wasn't made of card either.
(2) I got my first American Express green card (while I was still a student) by signing a document and returning it to them. A gold card followed a year later on exactly the same basis. I went in a PC-parts shop after the green one arrived and presented it to pay for whatever it was I bought. The guy treated it exactly the same as if I had presented him a week-dead fish.