Push to Upstream.. pushes your branch as you would expect for a remote-tracking branch.
Push… opens a dialog and lets you specify which refspecs you should push up, including tags, special cases like HEAD:refs/heads/master, etc.
That it doesn’t update origin/master seems like a bug to me, or maybe they’re waiting for the next fetch to update origin. I’d open a bug at https://bugs.eclipse.org/bugs/enter_bug.cgi?product=EGit&component=UI
PW