Is that strictly the definition, that the bit-width of a processor refers only to its native pointer size?
I know it's hardly a typical or modern example, but the N64 had just 4MB of memory (8MB with the expansion pack). It most certainly didn't need 64-bit pointers to address that pittance, so it was a "64 bit processor" largely for the purposes of register/data size.
Not so much a definition as where the problem lies.
If the thing that can refer to a memory address changes size, there are very different problems than will arise if the size of "an integer" changes.
You could easily imagine a processor that can only address an N-bit address space, but can trivially do arithmetic on N*M bit integers or floating point values. And obviously the other way around, too.
In general, I think "N bit processor" tends to refer to the data type sizing, but since those primitive data types will tend to fit into the same registers that are used to hold pointers, it ends up describing addressing too.
I know it's hardly a typical or modern example, but the N64 had just 4MB of memory (8MB with the expansion pack). It most certainly didn't need 64-bit pointers to address that pittance, so it was a "64 bit processor" largely for the purposes of register/data size.