When you login, you have a "session". You access files in the home directory of your account. GUI session has many processes that access files. If you did the previous commands from session of regular account, then the /home was in use.
The errors told that too: "contains a filesystem in use". The "umount" cannot disconnect the filesystem in /home, because something still does access files there. You did try to saw the branch that you were sitting on.
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LV VG Attr LSize Pool Origin Data% Meta% Move Log Cpy%Sync Convert
home cs -wi-ao---- <39.85g
That shows attributes of the LV:
-wi-ao----
o -- open. Filesystem within LV is in use. The 'umount' failed to unmount.
a -- active. The LV is in "active" state. That can be toggled with
lvchange when LV is not open. See
man lvchange
lvremove only LVs that are not open and not active.
You must check/know the state of your system both before and after you give a command.
Before, so you know which commands you need to give. You have shown that you repeat commands that you shouldn't.
After, so you know whether the command did have the desired effect.
Ctrl-Alt-F2 should change view from the GUI login dialog to text-based "virtual console".
There you can type username and password and then you have CLI session. Much less disk access.
However, you don't want to login with your regular account. It does not have home directory.
GUI login as root is disabled by default. You should be able to login as "root" in the console.
Account "root" does not have its home directory within /home. Therefore, it is safer to handle /home as root.