How to confirm external drive is being included in Time Machine backup

By default Time Machine backs up all mounted volumes when it runs, other than the TM backup disk. Your screenshot shows TM excluded it automatically, as it should. In other words everything looks correct.

TM is not intended to be used with third party file management or backup tools, so don't use them. The only way to confirm your external hard disk drive's contents are being backed up is to restore an item according to Restore items backed up with Time Machine on Mac - Apple Support. You should be able to scroll "back in time" to view older versions of files whose contents have changed. If you can do that, then it's working.

Browsing TM backups using anything other than the Time Machine interface is not supported. Altering them can corrupt the backup, making it useless.

Never mind, I figured it out ...

Providing confirmation things are working the way they're supposed to be working would be contrary to Apple's design philosophy, which is basically to stay quiet and to not get in the user's way. Unlike Windows (for example) which is constantly drawing attention to itself: "Look at me!!!"

If TM needs attention, it will let you know.

I have a similar question - I am pretty sure that I was backing up my external hard drive, which died two days ago. I am having trouble finding it as I browse my Time Machine, as there is no current file for it. How did you get yours to show up in the Time Machine interface?