When Intel was developing Itanium they named the new architecture IA-64 and retroactively named their 32-bit x86 line IA-32.
After AMD released "AMD64" Intel started copying the AMD 64-bit extensions. IA-32e was a short lived name and Intel started using "Intel 64" to refer to the 64-bit extension of x86 and continued using IA-64 for the Itanium VLIW CPU line.
https://en.wikipedia.org/wiki/X86-64#History_2
After several years of denying its existence, Intel announced at the February 2004 IDF that the project was indeed underway. Intel's chairman at the time, Craig Barrett, admitted that this was one of their worst-kept secrets.
Intel's name for this instruction set has changed several times. The name used at the IDF was CT (presumably[original research?] for Clackamas Technology, another codename from an Oregon river); within weeks they began referring to it as IA-32e (for IA-32 extensions) and in March 2004 unveiled the "official" name EM64T (Extended Memory 64 Technology). In late 2006 Intel began instead using the name Intel 64 for its implementation, paralleling AMD's use of the name AMD64.