I'd like at least two photo libraries that sync and are always the same. One on my Mac mini Ventura 13.3, one on icloud and one more on an external hd that is usually plugged in. Can you tell me if there are specific settings for this? Thank you.

You could achieve this by setting up iCloud Photos. iCloud Photos will sync your System Photos Library with iCloud and store the complete library in iCloud (minus your projects). And it will automatically keep the system Photos Library as a local copy of your iCloud Photos Library on your Mac, fully in sync with the iCloud version.

However, each user can have only one library syncing with iCloud. To get a second copy of the library on an external drive you will need to set up a second user account on your Mac. Set this user account up to use the same AppleID with iCloud and create the System Photos Library for this account on your external volume.

Then you will have a second copy of the library in sync with iCloud. I am simply using two different macs to sync with my iCloud Photos Library and to back up the library with Time Machine.

One way to have two local libraries on different volumes be synced is to use Carbon Copy Cloner. It is an incremental backup/sync app (it copies only those files that are new or have been modified) that will keep Library B exactly the same as Library A either on a set daily schedule, manually or whenever it detects a change in A it'll sync it with B.

I had mixed result with Carbon Copy Cloner on photo libraries. I stopped using it for the drives with photo libraries a few years ago, because I occasionally could not open the libraries again after syncing the drives with CCC. I suspect, that Apple's caution about storing libraries on synced volumes might also apply to volumes synced with CCC. Have you ever encountered problems with using Carbon Copy Cloner on photo libraries?

If we want an incremental backup copy for Photo Libraries we could use Power Photos to merge the changes from the library on the in internal volume to the library on the external, however.

You're welcome 😊