When compiling code with VC++, MSDN gives you the option between using the x86_amd64 toolset or the amd64 toolset (when calling vcvarsall.bat).
How do I choose between those two when compile x64 code? Will the amd64 option churn out more efficient x64 machine code than the cross compiler?
It has nothing to do with efficiency. The native and cross-compiler will both generate the same machine code. You will however gain some benefits by running a native 64-bit compiler process on a 64-bit workstation (larger registers, larger memory space, etc...).
The native compiler will only run on an 64-bit copy of Windows, so if your workstation is 32-bit this compiler won't even run.
The cross-compiler is meant to run on x86 machines even though it will run on a 64-bit copy of Windows via WoW; however, there is no reason to do this.
The page you link says it quite well:
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.Thanks to Brian R. Bondy for the quote formatting