I am downloading an ISO image of Lubuntu; they have two versions: 32 and 64.
But why do they call the 64 version amd64 since they say that it works for Intel also?
Because AMD was the first one to release 64-bit x86 (x86-64) CPUs.
the AMD64 architecture was positioned by AMD from the beginning as an evolutionary way to add 64-bit computing capabilities to the existing x86 architecture, as opposed to Intel's approach of creating an entirely new 64-bit architecture with IA-64.
The first AMD64-based processor, the Opteron, was released in April 2003.
In fact, in the kernel the 64-bit support is called 'x86_64' to refer to the fact that both AMD and Intel (and others) implement those instructions.
AMD intially implemented x86-64, before Intel. For this reason, most distros (and other OSs such as OpenBSD) did a first release when it was still AMD only, or renaming to x86-64 wasn't considered worth the effort.
I think one of the another contributing factors to this naming confusion was a conflict between Intel and AMD over naming. Intel had an identity crisis in it's 64-bit naming. It cycled through the names EM64T, IA-32e, and finally the official Intel 64. AMD on the other hand always called it AMD64. x86-64 was a fairly obvious name to cover both. However, it wasn't always like this. I remember when it was new and shiny people couldn't figure out what to call it. I've seen these:
x86-64 is the obvious best short name for the architecture