What are the time machine snapshot files that are saved on my local volume instead of the time machine back-up volume
Those are APFS local snapshots that use copy on write, so they take no space overhead to a first approximation. 

I wouldn’t even bother removing them unless you have measurable file extent fragmentation since you can roll back changes like software updates with these as well as the system cleans them automatically if you run into storage pressure and the external drive connects to get a second backup after a few days time. 

If you want to force one backup and then force deletion of all the snapshots here are the commands. I would be sure the first one completes without error, though.

tmutil startbackup --block
tmutil listlocalsnapshotdates / | grep -v dates | xargs -I date tmutil deletelocalsnapshots date


Note: the tmutil startbackup reports success on the command line (exit 0) no matter what, so be sure you see actual data copies and not zero values and a pop up error in notifications if you are logged in to the Mac console.