yep, that's right. Or rather, origin and master are just two parameters given to pull/fetch/push to describe a target while origin/master is just the local name

yep, that's right. Or rather, origin and master are just two parameters given to pull/fetch/push to describe a target while origin/master is just the local name for, as you say, the locally cached ref. Comparing against that locally cached ref is also what git uses to tell you how far behind/ahead of the upstream you are in `git status` or whatever. Fetch and push are the only git commands that actually talk to a remote (at the "user level" of the command set anyways, those are also composed of lower level commands).