I have a Macbook Air that has 2 partitions on the hard drive. One is for Yosemite (450GB) named "Macintosh HD" and the other already has Mojave (50GB) on it named "Macintosh HD 2".
When upgrading the Mojave to Catalina, I wanted to back up the Mojave partition ("Macintosh HD 2") and upgrade it.
But when I plugged in an external hard drive named "Seagate2TB", Time Machine started to back up hard drive 1 first (450GB) "Macintosh HD". Since I already set and boot up using Mojave, I restarted the machine, and ejected the 450GB partition "Macintosh HD", but when Time Machine started again, it still was backing up some 410GB of data, which means "Macintosh HD" is included.
So I went to Time Machine's Preferences, and clicked on Options, "+", and chose what to exclude: "Macintosh HD".
Next I rebooted the Macbook Air and wait for the back up to start, and it again, reported backing up 410GB of data.
How can "Macintosh HD" be excluded?
Best Answer
Your problem is a consequence of an over simplified vision of users from Apple: a user has one uniq computer with one uniq partition then name it with a simple and uniq name "Macintosh HD".
Once you start to have external disks, different versions of OS on the same disk, you should carefully avoid all the failures this over simplistic naming scheme will led you to.
Change all your partitions names with a very clear and really unambiguous name for you and forever.
My personnal use is as follows: every disk partition is named from the computer name it is used on ex.:
coriolis, the partition number ex.:1, its actual capacity, ex.:250 Gband the version of the OS it is used on, ex.:High Sierrafor a production OS or10.14.1for a test one.Ex.:
All my computers, all my external disks have a different and uniq name registered at time of their 1st formatting.