The problem is correct as stated, and solutions above by RichardKennaway and Oscar_Cunningham are correct. I think you may have missed that the prisoners are all distinguishable, a.k.a. they are numbered 1,2,3,.... Or you are confused about the win condition; we don't have to guarantee that any particular prisoner guesses correctly, just that only finitely many guess incorrectly.
Sub-puzzle: prove definitively that if the prisoners are not distinguishable, then there is no winning strategy.
In the winning strategy, do fewer than half the prisoners guess wrong? Do more prisoners guess correctly than incorrectly? I'm trying to get a handle on whether it is worth my while to try to penetrate the jargon in the "correct solutions."
If it's worth saying, but not worth its own post (even in Discussion), then it goes here.