amd64 is the original name of the instruction set. Intel did beat AMD to a 64-bit instruction set: that of the Itanium processors, IA-64. Itanium had performance issues and lots of errata. Most importantly, IA-64 was not natively backwards-compatible with x86 instructions. amd64 became the standard.
x86_64 is a common name for the amd64 architecture, and is a way to describe both the AMD and Intel implementations. In my opinion, amd64 is a less ambiguous name and is more historically accurate.
https://en.m.wikipedia.org/wiki/X86-64
Yes, I am aware that my point is undercut by the fact that the article title is x86-64, but I stand by my statement.