The number of bits on a CPU can be open to interpretation sometimes, but in most cases it is pretty clear as the largest word size the CPU can work with using the majority of its instruction set. So an Intel i7 with AVX512 is not a 512bit CPU because all the other standard ALU operations are still 64bit. So the ability to work with 512bit is just a special case limited to some special instructions.
Yes, this is a relatively common topic and has already been discussed here from what I remember.
The reasonable answer, of course, for a definitive and general definition of a n-bit CPU, is that there is no such formal definition. It's basically up to the designers of the CPU, and has sometimes been used as a marketing tool.
That said, I agree that the most common use of this term is neither for the data bus width nor for the address bus width, but for the width of the general-purpose registers of a given CPU (and, usually again, the width of the ALU inputs). Of course that applies to... the most common architectures. There are some odd architectures out there (there aren't now as many as there used to be in the past) that have different sets of registers with different widths for different internal uses, so in this case it becomes hard to properly define "n", but in many cases it's relatively obvious. As you said, registers used for vectored operations usually don't count for this; they are not general-purpose and can be used only for a restricted number of uses. But you may still debate "n". OTOH, on a classic RISC processor with no vectored operations, registers usually all have the same width and it becomes relatively non-ambiguous.
And there are a lot of examples of CPUs with various data and address bus widths not related to "n". The 8088, 68000, 68008... for the data bus (and probably many more), and still most processors these days regarding the address bus. The 8088 was considered a 16-bit processor (with an 8-bit data bus), the 68000 a 32-bit processor with a 16-bit data bus, the 68008 a 32-bit processor with a 8-bit data bus (externally, but still 16-bit internally?), and neither had an address bus with a width corresponding to "n".