In my case, no one "signed" for the aircraft as such. It's my plane (well, a proportion of it), I filled in the plane's flying record book, naming myself as captain. And to complicate matters, when I was checked out on a PA28-140, the plane's owner sat beside me, wasn't a QFI, and was happy simply to see I was reasonably proficient. I logged this time as P1.
So, being checked out on two different variants of type, means that, due to the presence of an instructor in one, different hours are logged . . . .