I don't know about "most".
My newer lathe is the 3 shaft design and also has a gear rack that runs the length of the bed.
My old lathe is not a mini lathe by any stretch of the imagination. I'd guess it's bed is 6ft long. It is just old. It has only one shaft and no rack. The one shaft has no keyway slot. Power Feed is done with the half nut. The half nut is sloppy as hell but it has not changed since I got it. I did some amazing work on that old girl. She taught me most of what I know.
I don't think this is an either or situation. I think they come in all kinds somewhat loosely depending on vintage, cost and size.
I have not been with enough different lathes to know how most are built. I like em all!