Trying to backup Photos to iCloud filled up my external drive with a “snapshot,” I think?

Just so you know the iCloud Photos feature is not an independent, off-computer, storage service but a syncing service. Any change you make to your local Photos library will be reflected in the iCloud Library, i.e. additions, edits, and deletions (both intended and accidental).

A backup plan that many subscribe to here is two types of independent backups:

1 - an external drive that's 2 ½ - 3 times the size of the drive(s) you're backing up setup with Time Machine. This makes an incremental backup every hour so if a file is deleted accidentally or you need a version from a day or two ago you can restore it.

2 - an external SSD that has your boot drive cloned to it (I use Carbon Copy Cloner for the cloning and backups). This lets you have a bootable replacement in the event your internal boot drive has problems. However, versioning is not part of this type of backup, i.e. you only get the latest version of all files.

Do you also have a similar size library in iCloud - but need to sync your local library with that?

If so, then your mac needs to download what is in iCloud, so it can compare with your local library. If that is what is happening, then the only solution I know of will be to get (or borrow) a large enough external drive.

You’re not using the same drive for time machine are you?