SSD external HD set as boot drive but desktop still the internal HD

Restart holding the Option (Alt) key and select the external SSD drive in the Startup Manager.

see > Change your Mac startup disk - Apple Support

If you only see the internal drive, then you did not make a properly external startup drive.

see > Use an external SSD as your startup disk … - Apple Community

Another test to do is click on the Finder app in the Dock and look at the bottom of the window that comes up to see where you're at. I've set the Finder window to open my Home folder and this screenshot is for my internal drive:

When I boot into my external drive and system the Toad Hall changes to what I've named it.

Hmmm, i have no trouble booting off external drives after using CCC here.

We need to see what all is running, a report from this will not display any personal info...

Using EtreCheck - Apple Community

EtreCheck is a FREE simple little diagnostic tool to display the important details of your system configuration and allow you to copy that information to the Clipboard. It is meant to be used with Apple Support Communities to help people help you with your Mac. It will not display any personal info.

https://www.etrecheck.com/

Thanks for Old Toad’s etrecheck instructions…

Slow iMac 2017 - Apple Community

Use the Note tool on the bottom of this editor's toolbar, as shown in the image, to copy and paste the output from EtreCheck. In a Reply before you click post, look for this to add longer texts...

Boot into Safe Mode according to How to use safe mode on your Mac on the external system and reset the default boot drive. Reboot normally and test.

Note:  if you have a wireless keyboard with rechargeable batteries connect it with its charging cable before booting into Safe Mode.  This makes it act as a wired keyboard as will assure a successful boot into Safe Mode.

A cloned drive will show the same desktop as the drive it was cloned from until you make changes.

Hmm... that is not the expected behavior.

The only other idea that I can think of, is to startup in Recovery Mode > go to Disk Utility > select the external SSD drive and run First Aid on it.

You're welcome. That is good news.

Normally setting the startup drive to external in both the internal and external system preferences does the trick.

If not, then you may need to backup to another external drive and.... erase the internal drive.

Sounds like maybe the external SSD is being missed during the power on and initial startup cycle.

Another option, is to erase the internal drive and then use the internal drive for non-bootable CarbonCopyCloner backups of the external startup SSD.

Do you want to change the Desktop picture on the external system? If so boot into that system, open System Preferences, go to Wallpaper/Desktop Picture and select a new image for your Desktop.

Changing the desktop back ground on one of the startup disks, is a quick and easy way to tell which disk the iMac is started up from.

That is a good sign. Does it hold for a normal restart?

ric_e wrote:

I do use Time Machine to backup to another external hard drive but I am hesitant to erase the internal drive.

I would be too...

Sounds like I don’t have much choice in the matter.

One is to just keep using the Startup Manager.