Which is a better implementation of a stack data structure (based on an array)?
push() and pop() have an asymmetry that should be reconciled.  push() does not indicate any thing special when the stack is full and a push is attempted.  It silently does not do the push.  Yet when an pop of an emty stack occurs, a NULL is returned (versus a pointer to int) allowing the calling routine to take evasive action.  A symmetric solution would both do something special - or not - on full/empty.
A function to inspect the top of stack, without popping, could be added.  This IMO would be useful.  Additional routines to report current stack depth or stack space left are something to consider.
Using unsigned int is OK, especially if the fixed array size is known to always be less than UINT_MAX.  I favor size_t for array indexing as it has better portability for large arrays.
A way to deal with popping or inspecting an empty stack is, as part of initialize(), to pass an int as the bottom-of-stack.  This value does not actually ever get "popped off".
The names of the functions are too generic ("initialize") and could well collide with other code.  Suggest something like Stack_initialize(), and Stack_push().
The OO part of me wants to see a copy and destructor function too.  The destructor could simple do count = max = 0;