How to use internal drive to..

First of all, cloud storage is not a backup. Unless it's specifically designed to be, which iCloud is not.
Second, I don't actually fully understand how using your internal hard drive as backup location would affect iCloud photo sharing.

 

I use Photos to download (each sync) all pictures from cameras, phones, iPad Pro, etc (library can be on what ever drive you wish)

So all pictures are in one library, well 3 libraries in my case - 1)Work, 2) Hiking, 3) Family and Personal

I have almost 20 years of photos so back up is important for me and I use Carbon Copy Cloner CCC to back up weekly to 2 separate drives several days apart - just in case a back up goes wrong. I can revert the other back up.

CCC is great to set up scheduled back ups and if the external drive is not connected (in my case it sussually isn't) then CCC sends an alert to connect the drive and it automatically starts the back up as soon as the drive is connected

Carbon Copy Cloner is worth every cent for back up! IMHO

 

Questions:

How large is the external SSD?
How much space is currently being used on it?

How large is the internal SSD?
How much space is currently being used on it?

My best advice:
Use either CarbonCopyCloner or SuperDuper to clone the contents of the SSD to the internal drive.

Now the INTERNAL drive becomes "your backup" -- because your PRIMARY drive is the external SSD.

If anything ever goes wrong with the SSD, with a cloned bootable backup on the internal drive, you can be "back up and running again in about 90 seconds.

 

"Backup" means there's an original somewhere as well. Do you have them on your external boot drive?

If you're using iCloud Photos currently, the cloud is your primary storage, not a backup. If Photos > Preferences > iCloud > Download Originals to this Mac is selected, then you could consider the Photos library stored on your HD to be a "backup." If "Optimize Storage" is selected, you do not have locally-stored full-quality copies of all your images.

To withdraw from iCloud Photos you first need to be sure full-quality images are on your HD. They will already be there if you're set to "Download Originals to this Mac." Just turn off iCloud Photos.

If you're currently set to "Optimize Storage," switch to "Download Originals to this Mac" and then wait until Photos finishes updating. Then you can turn off iCloud Photos on the Mac.

That will leave you with one copy of your photos on your HD. To have a proper backup, you'll need to back that up with your favorite backup method (Time Machine, Carbon Copy Cloner, a cloud-based backup service...).

If you aren't already making a regular backup of the entire external boot drive, do it! External drives are more susceptible to file system errors than internal drives due to the fact that you're dependent on that thin, exposed external wire - if the wire became even momentarily disconnected during a write, you may end up with file damage.

Most folks wouldn't consider an internal drive as a "backup" as accessing that drive should its "enclosure" fail can be difficult. Externals are preferred as they can easily be moved to another computer.

 

It looks to me like the image in the post directly above this one is what one sees when one mounts a drive with the Catalina OS on a Mac running Mojave...

 

You can't just erase "one" of the volumes if you installed Catalina and didn't like it.

You need to erase the entire drive and start over.

Mount the drive (with the two icons) on the desktop
Open disk utility
Go to the view menu and choose SHOW ALL DEVICES
This shows you the physical drive you need to erase (topmost line).
Now, erase that and start over.

 

This is my current scheme. The Photos Library.photoslibrary file gets backed up to Time Machine and also, once a month, to another drive that gets kept offsite.