Back up Photos Library which is on external drive (not in Pictures on iMac)

The easiest backup is to simply drag the Photos Library to another drive-- then you have two independent copies. But this only works if you have "Download Originals" checked in Photos>Settings>iCloud. If you have "Optimize Storage" checked, then your Library may not have all the original image files. I do this once a month or so.

I also use Time Machine to run hourly backups when my laptop is at home. Since your Library is on an external drive, your Time Machine Settings may be set to exclude external drives, so you would need to check on that.

Time Machine requires its own separate volume-- it shouldn't be on a volume with anything else.

Also, how do you keep the multiple copies updated?

Use an app that will do incremental back ups. The first time you run the back up the app will make a complete copy of the Library. Thereafter it will update the back up with the changes you have made. That makes subsequent back ups much faster. Many of these apps also have scheduling capabilities: So set it up and it will do the back up automatically. Example of such apps: Chronosync - but there are many others. Search on the App Store.

Is it better to have the system library in Pictures?

Makes no difference.

Also Time Machine does back up my external drives.

If you set it to.

OK-- I fear that there are some problem files that are not being recognized-- essentially a corruption of the Library. I imagine that you can still use it from Photos where it is as long as you don't chance on those particular files. It could take a long time to figure out which ones are the problem pictures. You can stop the rebuild and start it again-- I've seen it work better with a second try.

But, here's another idea. Since all your pictures are in iCloud, they are OK there. So instead of copying the Library you have that seems corrupted, you can create a new Library and populate it with the pristine pictures already in iCloud. The only downside to this is that it may take a week to finish.

So--create a new Library, probably on the new hard drive. (Or, it could be created anywhere, and then you'd just copy it to where you want it.) You make a new Library by closing Photos and then option-clicking on the Photos icon. You get a dialog showing current Libraries and offering to switch or create a new one.

The dialog shows what your current System Library is, and you can find the path of each of your Libraries. I have a bunch.

You create a new Library where you want it, and then make it your System Library. That's on option in Photos>Settings>General

"Use as System Library" is grayed out for me, because that Library is already my System Library. The System Library is the only one that syncs with iCloud, and it's the one that pops up for email, messages, and so on.

Then connect this Library to iCloud:

Select "Download Originals," since this is a backup. ("Optimize" loads only smaller versions of the images.)

Then sit back and let Photos copy all the pictures from iCloud to your new Library. Could take a week. Probably you should leave the computer on with Photos running in the background-- a lot can happen at night.

When it's through, you can switch back to the other Library (if it works,) or you can copy this one and use it. You will have to tell Photos which Library you want to use as your System Library-- there can be only one.

This new Library should be pure and un-corrupted, but it probably won't have the pictures or videos that caused problems with the other one-- or it may have them from before corruption happened. I don't think I'd erase the original until I was sure it was of no use.

Let me know!

I didn't ask, but how are your drives formatted? They must be formatted in either APFS format or Mac OS Extended (Journaled) format. Additionally, the drive can not have had Time Machine on it since it was formatted. Photos Libraries can be copied to other formats but they won't work and might get corrupted.

Move your Photos library to save space on your Mac - Apple Support

The advantage in keeping the System Library in the Pictures folder is that the computer's internal drive is way faster than any external drive. So you keep it there if you can.

You can have multiple Libraries. I have a Library for pictures of my wife's ancestors, one of my family's ancestors, one of work pictures, one of pictures taken with my Nikon, and one with pictures taken with my iPhone. (And I have others.) The System Library that's connected to iCloud has just the pictures that I have dubbed "favorites." These are pictures I may want to share with others, so I don't want ugly pictures there. I copy the favorite pictures from the other Libraries, and I copy the iPhone pictures from the System Library to archive them in my iPictures Library. As you can imagine, many of these Libraries don't change much.

Many of us use the 3rd Party app  PowerPhotos ($30) that helps copy stuff between Libraries, and it makes switching Libraries really easy. It also has some powerful and helpful tools for handling Libraries. So that's how I keep the Libraries straight.

The error -36 is usually a copy error when copying between volumes with a different filesystem format or different access rights.

If your two drives are having different file system formats it may not be possible at all to copy the Photos Library as an entity. Then you either have to use cloning software like carbon copy cloner or copy the library in parts. Open the package to show the package contents and copy each subfolder separately and skip the items that cannot be copied. This is described here: How to avoid the Error -36 when copying a Photos Library - Apple Community

Just drag the Library to the Pictures folder. Then tell Photos that this new Library is now your System Library. It's OK to have multiple Libraries (I have a bunch), but the only one that syncs with iCloud is the System Library. You designate the System Library in Photos>Settings>General. The Pictures folder is a good place for the Library, because the internal drive is faster than external drives. You should keep at least 15% of your internal drive free-- the OS swaps stuff in and out of there and it needs the room.

If you close Photos and then option-click on the Photos icon, you can see a list of Photos Libraries that Photos knows about, and you can switch between them. (Or you can double click on the Library you want to see.) It also shows which is the System Library.

The Library is "package," which is just a folder with some restrictions. It holds picture files and a database that connects the pictures together with metadata and edits. It works like other folders, but in using it Photos needs the ability to do lots of background work, so the Library can't be used on a non-mac [ APFS or Mac OS Extended (Journaled) ] drive-- the database can be messed up, otherwise. And you shouldn't mess around inside the package, because Photos may get permanently confused.

Your hard drives are fine. I've copied between APFS and Extended and not had problems-- they are both newer Mac formats. APFS is faster on SSDs, but maybe not on mechanical drives. Big problems do occur with older and non-mac formats that don't support large files and long names.

Something else to check, but is usually OK, is the permissions for the drive volume-- just be sure that "Ignore ownership" is checked in the drive info window. It usually is.

Since you've been syncing with iCloud, trust that and get a clean copy of your Library filled by iCloud. Then see if anything's missing, and you can examine the old Library more carefully if you need to.

Well, drat-- it's always somethin'!

Try these things, probably in this order:

• Restart the computer (of course)
• Re-start in Safe Mode. This bypasses certain potentially disruptive processes. Safe Mode is different for different computers, so see this: 

           Start up your Mac in safe mode - Apple Support 

• Rebuild your Photos Library--close Photos and re-launch by option-command-clicking the app icon
• Make a new user and see if the same problem recurs with it
• Use another library if you have one, or make a small test Library with a few pictures in it and see if the same thing happens. You can create a new Library by closing Photos and then option-clicking on the Photos icon.

I hold the most hope for Safe-Start, and for Rebuild. I doubt that it's the last two. Restart is very easy, Safe-start is fairly easy, and it clears up little glitches and prevents some processes. If the Library is slightly corrupted, the rebuilding will help. Good Luck...