What does Android rooting imply in terms of user accounts?
As you might have expected from the "Linux user perspective", item #3 is the correct answer. With a little adjustments: apps that come with "root features" can use the "root functionality" as well. When they try it the first time, you'll have to approve that, though:

Basically, a "clean rooting" brings in two things: a su binary and a kind of "supervisor app" (SuperUser/SuperSU). The former you can use e.g. at the shell to obtain root access – and the latter controls whom superuser powers are granted. The first time an app tries to obtain those powers, you will see a popup on the device asking you whether you confirm or deny. A check-box lets you store your decision permanently, if you wish; leaving that unchecked, the popup will re-appear the next time.