I still shake my head at how they were able to successfully rebrand it from amd64 to x86-64; yet ia32 is still semi-accepted as the "official" name of x86.
It's the opposite: the original name was x86-64, and amd64 is a later rebranding. See, for instance, the original web site for this (then) new architecture: https://web.archive.org/web/20000829042251/http://www.x86-64...
That's partly because Microsoft had support for ia32 (x86) and ia64 (Itanium). So amd64 was to differentiate the architecture (and it was unclear Intel would ever extend x86).
The "64-bit version of x86" is actually the new ISA created by AMD that was backwards compatible with x86 and added a 64-bit mode.
I'm sure that you agree that the company that creates a ISA has the right to name it.
I read this somewhere a couple weeks back, but I don't remember the source.