- Aarch64, the execution state
- A64, the instruction set
- ARMvX, the vX of the architecture, both of the above were introduced in ARMv8
ARM64 is kind of a loose umbrella term that might or might not have been officially retconned.
Why is that? Because ARM specifications are modular, so for each vX there are mandatory and optional set of features.
There is no official name for the 64-bit instruction set for the x86 architecture, as far as I know. As far as I remember, AMD called this ISA "AMD64" when it was released and Intel just called it "Intel 64". Of course, this term only refers to the basic instruction set. If you see an x86-64 binary, it may be compiled with an extended instruction set like SEE4 or AVX that is not necessarily supported by every x86-64 CPU out there.
Even the umbrella term "x86" for the 32-bit ISA was retconned as far as I know: Intel did not use that term originally.
This is to distinguish it to the for a long time common 32 bit CPU architectures.
Looking at the current market you could think that there are mostly 64 bit CPUs sold nowadays, but many people still use (older) 32 bit CPUs.