It's more than confusing you are right.
Your version 13.0.1 is CK1 (College Kings) the first game. It covered Saison 1 ACT1 to ACT4 and it finished at version 15.1.0.
After that they made the Saison 2 as a new game starting at the of S1 ACT4 but was labeled as CK2 ACT1.
Colleg Kings 2 started at version 3.0.0 (I think) and it's previous last version was v3.0.16s.
Now, they made change (to fix bugs and to avoid confusing people with the naming convention...), so the last version of CK2 became College Kings 2 Act 1 v0.9.1.s
So, if you are on version CK1 v13.0.1 you should get the last CK1 version which is 15.1.0 (and play it until the end to be sure to see all content).
Then if you want to continue on the plot, you will need to play CK2.
Keep in mind that actually your saves game from CK1 can't be imported to CK2, it's buggy as hell.
What you can do is to start which the path builder and choose the options to reflect your CK1 game play.
I didn't played with the CK2 v0.9.1s yet, only CK2 v3.0.16s, so I'm not able to discuss about it's content.