In git lingo origin is just the default name for a remote from which a repo was originally cloned. It might equally have been called source or remote1 or just remote.
Remember that git is a peer-to-peer, distributed system, not one with any built-in notion of client/server, master/slave, parent/child relationships (though these might be imposed upon it by a user in a particular scenario).
All remotes are equal. origin is simply (and literally) the first among those equals (for a cloned repo). 🙂
And as Jan points out in the comments, the name associated with each remote is intended for your convenience. If you find that origin does not really work for you then you can change it.
As for your interpretations of the push statement, your first is the closest to being correct but the push command as written will push the local master branch to the master branch on the remote identified by the (locally configured) name origin.
If there is no master branch in the remote then one will be created.
Full details of the push command and the flags, options etc are of course in the docs.
You rarely (if ever) refer to the ‘local’ repo explicitly since your operations are performed in the context of a repo.