Nah, only historically. Yes, "8-bit" refers to the ALU. Back in the day, 16-bit was similar. But even then it was muddy, because when talking about operating systems like Unix or NT the key question about bitness would be in the context of a 32-bit flat addressing model, not really the data width.
By the time the 64-bit era rolled around, the "64-bits" definitely referred to address space.. The original Pentium had a 64-bit data path and had instructions (MMX, eg PADDQ) that could operate on 64-bit numbers - no one would call it 64-bit. By the early 2000s with the big push to mainstreaming 64-bit, it was all breaking out of the 4GB address space limitation - not the width of data.
> define the bitness of CPUs by their capacity to add numbers in one go
This is nebulous and therefore troublesome to define. Are we talking about the ISA or the internal circuitry (ALU and/or data path)? Is the 68000 a 16-bit or 32-bit CPU?