Photos library on an external hard drive - keep it connected all the time?

As Mark wrote, after you sign into the user account, you have to make sure the external volume with the library is connected and available, before you launch Photos the first time and until you log off again. Otherwise the Photos might lose the connection the system photos library. For libraries that are not the system photos library and not syncing with iCloud it is not so important. See: Move your Photos library to save space on your Mac - Apple Support

Always make sure that your storage device is turned on and available before opening Photos. If Photos can't find the device, it stops using the Photos library stored there.

Bondia wrote:

If my Photos are synced with iCloud, they will still take up space on my Mac desktop, won't they? Or no? Can I sync an external hard drive to iCloud? And then eject the external hard drive and still access my Photos through iCloud.com? Is that what you are saying?

If you set your Photos>Preferences>iCloud to "Optimize Mac Storage," then only smaller versions of the pictures may be stored on the drive. The pictures look the same, and you seamlessly get full-sized versions when you edit or print pictures. This is great when your Photos Library is on a Mac with limited internal storage. The full-sized pictures will always be at iCloud.com. You can do this on iPhones and iPads, as well, to save storage.

If you have a Library on your Mac and one on an external drive, they can't both be synced with iCloud at the same time. Only the "System Library" is synced. You have full access to the non-synced library; you can edit, annotate, crop, delete, whatever, but it won't be copied to iCloud and your other devices. (You can swap the "System Library" designation back and forth, but you'll probably end up with lots of duplicates.)

You can always view your synced pictures on the web at iCloud.com, upload, download, etc. But it's clunkier and not as full-featured as the Photos app on the Mac. So, yes, you can "eject the external hard drive and still access [your] Photos through iCloud.com" even if you have no internal Library. If you make changes, then when you plug back in, the changes will be mirrored in you external Photos library.

Photos has many background processes that analyze and index the contents of your system photo library. It should remain available whenever your Mac is powered up, or at least wherever your Mac user account is signed in.

If you have an iMac, as shown in the lines at the bottom of your post, you may as well leave the external drive connected and powered up all of the time.

The library cannot sync to iCloud if the disk holding it is not connected to your Mac and powered up. How could it?

I just imported all images (36 000) and movies (3000) to a new Ventura Photos.app library. I chose to create the library to the internal disk because I first wanted Photos to detect faces (it took a few nights and required disabling sleep, I let the display turn off. Just keeping the Mac on by disabling the screensaver via a hot corner does not seem to be enough for the face detection to start).

When the face detection was done, I copied the library to a APFS external disk which I turn on only when I want to access the photos, or sync them to an iPad, or when detecting new faces in newly imported images.

Really, disconnecting the external drive is no problem. You just can't use Photos when it's not attached. But you can use iCloud.com to keep track of your pictures even when the drive isn't connected. If you have a laptop, for instance, you just plug the drive in at night, leave Photos open with another app up front, and all the background processes will happen while you sleep. Do that maybe once a week, or when things have changed.

Are you using Time Machine? You need a good backup, and TM is great. For a laptop, it's the same-- you just plug it in for the night, and it automatically backs up your machine. It can do both at the same time.

You're welcome. Good Luck for your library transition to a new drive.