What you can do without installing git on the server is maintaining a local copy of the rsync'ed server website on the local computer.
If your server supports sftp, you can use csync (without having to install it on the server side): it is better than rsync and bi-directional.
Then you can have your own git repository, and whenever you want to add or compare changes from a website, you can switch the branch and do:
git --work-tree=/path/to/csync/folder status .
Compare the current current repo index and the synchronized local folder that reflects the website.
You have two branches in your git repository, one for your current job and one for including the website in the repo.
You can combine the two locally, and when you are ready, you will check the repo using the csync'ed folder as a working tree
git --work-tree=/path/to/csync/folder checkout HEAD -- .
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