A couple of things: We're talking about all puppets here, be it famous puppets of stage and screen, puppet toylines, or even that special puppet you had as a child. Stop motion is allowed because it does meet a lot of the main criteria. However, as much as we love Big Bird and Sweetums, they are played by people in suits. So while they are Muppets, they do not count as puppets for the purposes of this list.
I feel like this rule makes a MUCH fuzzier line than you'd think. Aside from the obvious "well, any hand puppet is some percentage of a guy (a hand, probably some section of arm, possibly a whole second arm!) in a suit. Sometimes even two guys, like the Swedish Chef, but then you get into stuff like the whole style of puppet where you have someone in an all black suit controlling a puppet in front of them with rods hooked to various limbs, a full-bodied performance, but at most you have maybe a hand for the mouth actually inside the puppet. You're explicitly disqualifying Big Bird, but if you have him sitting behind a desk, we've got someone's hand going up a neck to his head, like a traditional hand puppet, and one of his arms being basically a glove is pretty common to other puppet designs. Similarly we have Ludo from Labyrinth, who as this test footage shows, is kinda straddling the whole intersection of suit actor/hand puppet/robot. Legs in wearing fuzzy pants, arms controlling much larger arms via some elaborate stick rigging I assume, remote operators controlling face movements with joysticks offstage.
Gordon "ALF" Shumway is usually a hand puppet, but in some scenes is a guy in a suit. R2-D2 is technically always a guy in a suit (or, was originally? I'm not actually sure in the new movies). I'd pretty confidently call the Faun from Pan's Labyrinth a guy in a suit, but if I had to strictly define how I draw the line I'd probably get into how directly limbs are controlled and turns out that's more of a corner case than I thought.
Snuffalupagus is another weird one, because I'm pretty sure there's a whole human being standing in there but generally only his head is doing anything, and full body stuff I'd assume requires another person for his back legs (and then there's also the whole Dragon Dance thing in a similar vein). And I want to say "all kaiju are people in suits" but I'm genuinely unsure about like, Mothra, or some appearances of Ghidorah. I think, tentatively, all my picks are unambiguous but... yeah I'm not actually 100% sure.
Another weird corner case is when you have say, Audrey II, who I think is technically a completely different puppet in every shot of the movie, but I have to assume the answer there is "one character=one puppet" because I mean how do you even count otherwise.