Big Sur mixing up file names on external hard drive

So last evening I dragged the external onto "Privacy" under Spotlight in order to rebuild spotlight. The external has never shown up on either of the spotlight options: "search results" or "privacy", and does no longer show up in my finder either. It does though show up under Disk Utility.

I have restarted the computer and left it like this all night. There is a lot of data on the external but, no change this morning. How do I get Spotlight to re-recognize my external so that I can ask it to rebuild? Many thanks!

It's formatted as Mac OS Extended Journaled. I ran Disk Utility on both my Mac and the external hard drive and I think for now its working but I had noticed the problem before. Seems to come and go a bit.

How do I turn off spotlight on Big Sur? Thanks

Ok, so I went under Finder Preferences and found that the external drive was now unchecked. I re-checked it but spotlight was still giving mixed up results. So I did the procedure once more, this time not dragging the external onto the "Privacy" list of Spotlight, but rather selecting it via the plus sign. It seems to be working. Thanks very much for your help!

The external was actually mounted in Disk Utility. (Sorry, my picture did not show that fact.)

But rechecking the external in Finder Preferences did the trick.

I think my initial error was to drag the external onto Spotlight's "Privacy". Something went wrong in that move.

Selecting it via the plus sign made it show up on the Privacy list, and allowed me to subsequently remove it from that list.

Thanks for your help!

Well, so I dragged the external drive onto the privacy side of the spotlight 3 hours ago. Since then it does not show up in my finder anymore. Now what do I do?