I'll try to break down into manageable chunks:
Old skool terminology for hard disc partitions went something like /dev/c0t1d0s0 (you don't wanna know! though some might figure it out - clue: scsi hard disc example ;-))
Then came 'hd' to denote hard disc, with the number corresponding to the partition. Primary partitions numbered 1st, followed by extended, followed by logical. Hence I can't guess at your actual partition numbers - depends if you have multiple primary partitions (as I do).
Latterly, 'sd' (standard disc?) started to be used but not by all distros/derivatives. I think this was to cater for USB devices - wikipedia anyone?
Most recently UUID is being used (I hate it!) - so ignore where I can
😉
That's disc naming conventions kinda explained?
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The name of the partition in GParted has no bearing on any other bit, mostly - just to identify whilst partitioning.
You have a Home partition (whatever you call it) and within this the contents of the home (lower case) directory.
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To give exact commands I need to know the layout of your disc partitions - which ones are Ubuntu and which Mint.
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gedit: I wasn't too clear. sorry - merged two lines of thought badly. All I meant was to add a new line of text to the end of the file.
Your spacing is gone to pot again!
/dev/sda3 is the 3rd partition (as an example) and should match the partition number of your new home partition.
/home is what is termed the mount point (where to find the files in the partition)
ext3 is your chosen file system type and don't worry about the other three parameters. Suffice to say there are 6 parameters, which should be separated by tabs - spaces will do but stick with tabs (convention).
So, if your new "Ubuntu Home" partition resides in the 3rd partition then you need to enter the following on its' own on the last line....
/dev/sda3 [Tab] /home [Tab] ext3 [Tab] reatime [Tab] 0 [Tab] 2 [Enter]
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If you use auto-login for Mint (as I do) then failsafe terminal mode won't be obvious. Otherwise, it's in the Sessions tab when your asked for your login details. If you use autologin, press Ctrl+Alt+Backspace to kill the X Windows and return to the login screen.
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chown:
When you copied the files over using the live-CD, the ownership of the files/directories in the new home will be wrong. We need to change ownership (chown).
If you normally login to Mint as a user called 'user' then you need to enter...
sudo^ chown^ -R^ user^ /home/user
If you normally login to Mint as a user called 'otherego' then you need to enter...
sudo^ chown^ -R^ otherego^ /home/otherego
(Once again a caret denoting spaces).
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Drat the G&T has evaporated! :_|

Just skimming the surface
Opinions/guidance expressed are intended to benefit the reader (mostly) but no responsibility should be assumed for the accuracy and no warranty is implied/expressed or given - so eBay may pull this post
Demised responsibility