Time Machine

I discovered Time Machine backups ("backups.backups") in the trash on my MacBook Pro (Retina, 15-inch, Mid 2014) running up-to-date Catalina. I deleted them, but now see that on the next Time Machine backup, they appear again. Additionally, there is another "backups.backups" on my hard drive in a folder named "com.apple.TimeMachine.localsnapshots" whose creation and modification dates are equal and from months ago. (I have automatic backups on and pointed (I think!) at a Time Capsule on my local network. Time Machine preferences shows the selected disk as my AirPort Time capsule name".local" .

There are two aspects of this that I would like to resolve. Most importantly, I want to make sure backups are being performed as expected, stored on the Time Capsule (which is shared by my iMac), and available for full or partial reload if necessary. I am untrusting of the existing backup at this point and think my preference would be to delete the entire thing and start with a fresh set. I have an external drive to which I have copied the most critical user files. I am fine with starting over with the iMac backups as well.

Secondly, I'd like to get rid of any unnecessary clutter on the internal hard drive of the Mack Book.