In general: Backups are always essential, regardless of what you do. A common word here says, that data, that are not backed-up are unimportant or even senseless data.
In case of great changes - as installing a new OS or OS version, backing up should always be the most crucial part.
Besides of that, I am astonished, that the old system worked at all. Seeing an Extended partition tells me, that you have an msdos-partition table (in difference to gpt). And in a msdos-table there can never be more than 4 primary partitions inclusive the extended partition, but your first picture shows 5 in total. (An extended partition is a special usage of a primary partition.) Further more a extended partition, can not contain any data - also no swap, as shown in the picture - directly, but only logical volumes.
Again, I am astonished how this could ever work. And the Mint installer seemed to wonder about the same question and corrected the wrong partition table.
One other thing that makes me wonder is the point, that partition 3 (previously Maya) has before and after the new install exactly the same amount of used space. That may be correct, but appears to me as really strange.
The third thing are the 2.26 GB data in /home after 10 minutes of installing. Maybe a leftover from the previous /home-partition, but I wonder, if the data is correctly stored. I would backup the whole machine and would make a complete new start.