Simple Python Code that zips file
You're using pathlib, which is great! Unfortunately, you immediately discard the functionality it gives you, and use old-style os APIs.
A good place to start is here - a mapping between the os[.path] APIs and their Path equivalents. Doing that gets us here:
def make_archive2(source, destination):
    shutil.make_archive(destination.stem, destination.suffix, source.parent, source.name)
    shutil.move(destination, &quot;/Users/my_name/Desktop/Zip Test/zipped_content&quot;)

We now don't need to do any manual manipulations using os or os.path. Once we're here, we can actually realize that the first parameter to make_archive is a full path to the file, completely removing the requirement to call shutil.move
From there, we can improve some of the other things you do. For example:

Your source and destination are the same thing... but you don't actually use the destination in a meaningful way
You extract the format from the destination, which feels a bit odd
You hardcode the actual final destination
You use both the root_dir and base_dir parameters, when that really isn't necessary
You don't return where the file ended up, which feels nicer than the caller having to set up their Path themselves

Altogether, I ended up with something like this:
def make_archive3(to_archive, move_to, archive_format=&quot;zip&quot;):
    move_to.mkdir(exist_ok=True)
    return shutil.make_archive(move_to / to_archive.name, archive_format, to_archive)