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" too.

When upgrading the Mojave to Catalina, I wanted to back up the Mojave partition ("Macintosh HD") and upgrade it.

But when I plugged in an external hard drive named "please fill me", 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?