I need a bit of help to see if I understood this correctly.
The internal register of a CPU is what is called L1 cache and on some/older CPU's L2 cache is also internal, but nowadays L2 is primarily an external register or memory between the CPU and RAM. Correct?
(for the sake of quriosity. Where is L2 placed if it's external? on the mobo inside the northbridge?)
Also, the actual size of the internal register has no other meaning for the processor other than it can store more data the bigger it is right? I mean, it has no impact on speed or anything else correct?
Er... L2 caches on on the same die as the procesor pretty much because they stopped being external back in the Pentium 2 days already...
Edit: What you decribe there does however in some aspects sound like the snoop cache, think unofficial L4 cache, which sits on the northbridge of all modern Intel systems and controls memory access cross multiple processsors and / or processor dies.
Essentially a register is a single byte (or word/dword/etc depending on architecture) section of memory in the CPU core that is acted upon directly by machine code instructions to perform operations (like add, increment, multiply etc), while any other memory is just that, whether it's on CPU cache or main RAM, and the only instructions that access it are ones that fetch data from it, or store it back.
To understand them properly you'd have to start off looking at something a lot more basic than modern PC CPUs, that have a lot of registers.
I'd recommend looking at some much older CPUs (Z80, 6502), or if you're after something current, PIC Microcontrollers (like the PIC16F84), as they have a very basic architecture that's easier to learn CPU structure from than something like a Core2.