Nine pawn cheating problem
There are many of those, some brilliant. But they are not just about pawns but with a board full of pieces where removing any unit leads to #1,
Oh nice, do you know some keyword to help me find those?
I think it can be ranked under the heading of illegal cluster though there are slight differences.
In the popular illegal cluster genre you first add the units on a given list to a legal position such that the position becomes illegaL - but legal after removing any unit on the board! The 9-pawn problem is an illegal cluster where the addition phase is skipped and only pawns feature in the removal phase. And there is the extra checkmate in 1.
https://www.ozproblems.com/walkabout/walkabout2022/sep29

Oh nice, do you know some keyword to help me find those?
I think it can be ranked under the heading of illegal cluster though there are slight differences.
Thanks for linking to my intro to illegal clusters! However, I don't think the OP problem relates to that retro type. The OP problem simply belongs the category of joke problems, because the solver is not informed of the 9 pawns. The solver has to notice this illegality and thus "gets the joke" on their own. Illegal clusters are of course a standard retro type where no information is hidden from the solver just for fun. And as you mentioned, the OP problem contains a different M1 for each removed pawn, and that's the vital theme not seen in illegal cluster problems.
Anyway, the Fischer composition I posted is the best-known demonstration of the theme. Since the OP says it's not the one they're looking for, I went to a database to see if there are other examples, and found a few. These other problems are not anticipated by the Fischer because in each case the 9 mate-in-1s are arranged differently, though the exact same theme means the later examples are not very original.
https://pdb.dieschwalbe.de/P0002015
https://pdb.dieschwalbe.de/P0005862
https://pdb.dieschwalbe.de/P0006015
https://pdb.dieschwalbe.de/P1268404
Thanks for linking to my intro to illegal clusters! However, I don't think the OP problem relates to that retro type. The OP problem simply belongs the category of joke problems,
That is true of course! What is also true is that there is a core phase in solving an actual illegal cluster - removing single units from an illegal position to show it becomes legal - which is very similar to the one in this joke problem type. I'd say they are "associated by concept" though there is a list of differences!