time machine backup file nam

Have you looked for Time Machine Snapshots on your internal drive?

About Time Machine local snapshots - Apple Support

If you enter Time Machine when your backup disk isn't available, all the backups that you can access are from local snapshots. You can restore from these backups ...

You can try to find Backups.backupdb on your external but I'm not sure it will do much good as Time Machine uses a specific file structure which is probably mince meat now from using a recovery software.

search for .backup files or just erase the time machine and start over with a fresh new back up. I also use a seperate hard drive and make a complete backup using Carbon Copy Cloner once a month in case the time machine every fails or in your case is stolen.

search for .backup files

Sorry, but that is not correct. Time Machine does not create files with the extension of ".backup" on local drives. Instead the "main" file is a folder, named: Backups.backupdb. Inside would be a folder with the Mac's computer name. If more than one Mac is backing up to this same drive, there would be a folder for each of them as well.

Time Machine uses backup folders for backups to locally-attached drive.

If they exist, they would be found in Finder Volat:

/Volumes/<drive name>/Backups.backupdb/<computer name>/