Well, I don't know if a categorisation would work, because the best puzzle
is an original puzzle. But as a start:
[Note: some examples include spoilers, especially for Theatre, since that
was the last major game I played.]
1 - The Generic Boring Sequence "Puzzle" aka Object Matching
Some hazard requires an object to pass. The object is hidden in
another area of the game. The object has no other usage, and once you
know about the existance of both the hazard and the object, the solution
is blindingly obvious.
Note: This isn't really a puzzle at all, but tends to be used to ensure
that the game is solved in a fixed order; hence Sequence Puzzle. These
puzzles are dull but may be necessary.. but please don't include too many!
Examples: Locked doors, or feedable monsters, are favourite.
2 - The Examine-Everything "Puzzle"
Something has a hidden object or similar, which is revealed by looking
at/under/on/around it in a certain way.
Note: This, IMHO, is worse than type 1 if badly done. Just having to look
at, around and under everything in sight is really dull. But if well done,
it can be rather good.
Bad Example (IMHO): The chest under the bed in "Balances". Apart from
the things-under-the-bed cliche there's no reason to do this.
Good Example (IMHO): The Secret Gallery door in "Theatre". A picture
is the sort of thing you're going to look at and doing so gives you just
enough of a clue.
3 - Usage of an object as something else with which it has something in
An innocent looking item is provided which has something in common with
another object that would be useful. You have to use one as the other.
(Curses) A dumbwaiter goes up and down between floors. So does a lift.
So, use the dumbwaiter as a lift.
(Theatre) A chandelier swings from the ceiling; so could a rope. So, use the
chandelier as a rope. A piano has a flat surface high up; so does a
stepladder. So, use the piano as a stepladder.
4 - Violation of a percieved internal restraint.
A puzzle which is solved by doing something which, although technically
possible irl, would not be the sort of thing one would actually do.. not
sure about this.
Example: The snake in the locker. Conventionally most people would think
to confront the snake somehow. In fact, the solution is to move the locker.
Although irl, most lockers could be moved, it is not an acceptable thing to
do, so quickly people develop a complex which means they don't immediately
think to do it. (Probably).
To be continued.. (maybe)