What is the relation between word size and size of internal register of a processor?
The term "word size" is defined in different ways by different sources, but almost everybody agrees that it is architecture dependent.

In the definition you quote: "number of bits processed by a computer's CPU in one go" it is not clear what "processed" refers to.

(Sorry about using ancient example, but I have not really looked at hardware specs for ages)

For example: The 486DX processor can add 32 bit integers for a 32-bit result (plus carry).  It can multiply 32 bit integers for a 64-bit result.  It can add or multiply 80-bit floating-point numbers for 80-bit results.

It has 32-bit integer registers and 80-bit floating-point registers.  The 64-bit int multiplication stores in result in two registers.

I won't even start on the various physical and virtual memory address sizes.

The integer multiply instruction 32x32 -> 64 is an oddity, most integer instructions can only process 32 bits.

Now, what is the word size?

It all depends on your definition.