If you have already cloned someone else's repo (rather than forking) and have been committing locally, what's the appropriate way to push that into a new remote repo on GitHub?
You can change the remote or add another one.
To change the remote
git remote set-url origin https://www.something.com/my_repo.git
To add a secondary remote as e.g.
git remote add other_origin https://www.something.com/my_repo.git
If you changed
origin you can
git push to the new remote. If you added another remote, you can specify during
git push which remote you want to push to.