amd64 and x64 is the same. You can use the amd64 on an Intel processor as well as on an AMD processor; there's no difference.
AMD, not Intel, first invented 64-bit instructions for Intels x86 processor architecture, and therefore sometimes it is called "amd64". Intel later implemented AMD's 64-bit instructions in their processors and called it "x64".
Note that there's also IA64, which is different. This is for Intel Itanium processors, which are very different from Intel x86 processors that you find in normal PCs.