From what you linked:
x64 on x86 (x64 cross-compiler)
Allows
you to create output files for x64.
This version of cl.exe runs as a
32-bit process, native on an x86
machine and under WOW64 on a 64-bit
Widows operating system.
x64 on x64
Allows you to create output
files for x64. This version of cl.exe
runs as a native process on an x64
machine.
Paraphrased:
If you use x86_amd64, then you are typically developing on an x86 machine and you want to create x64 files that run natively on x64. You could also use this option on an x64 machine but your compiler will be running under WOW64 emulation.
If you use AMD64, then you are developing on an x64 machine and you want to create x64 files that run natively on x64. The compiler is running natively in x64. This option is more efficient to build x64 programs.
You may wonder why you would ever develop an x64 program on an x86 computer, since you can't run it you can't debug it. Well it's still useful for example if you have a build server which is x86 and that build server needs to generate both x86 and x64 outputs.
How is it possible for a compiler to run under x64 if it is an x86 based program (x86_amd64)? That is the same reason you can run any x86 program on your x64 machine... Thanks to WOW64 emulation.
What is WOW64 emulation:
WOW64 emulation happens when you run an x86 program on an x64 computer (or IA64). WOW64 stands for Windows 32 on Windows 64. It is an emulation layer on top of x64 machines which allow you to execute x86 programs.
Your file system operations will be redirected to WOW64 folders and your registry will be redirected to a subnode as well. For example when you try to obtain the folder for program files it will return c:\program files (x86)\
if you are using WOW64 but it will return c:\program files\
if you are using x64.
Another example, for the registry if you try to write to HKLM\Software\Something
it will really redirect you to HKLM\SOFTWARE\Wow6432Node\Something
without your x86 program's knowledge.
Running a native x64 build will be more efficient than running through WOW64 emulation Why? Because you don't have that extra emulation layer of transforming your 32bit calls into 64bit ones.
By the way if you are running the x64 version of Windows you can see which processes are running through WOW64 because they will have a *32 appended to the process name in the process list.