In fact, the term "x86-64" came into existence as a term-of-art because "x64" technically refers to the Itanium ISA (Intel's first now-forgotten attempt at a 64-bit ISA), not the IA64 ISA (Intel's canonicalization of AMD's unofficial extension to IA32.)
The most correct term is "amd64" IMHO.
Uh, "IA-64" is Itanium. Intel officially calls x86-64 "Intel 64".
Yeah, Itanium is IA-64. It's also dying but not forgotten: people who like RISC, reliability, security features, and high-speed all in one remember their vain attempt well. Are also reaping benefits of used HP Integrity and SGI Altix boxes on eBay selling cheaper than consoles. I'm eyeballing a few of those suckers myself. :)