So yeah, finally got this working. Before switching to C3 runtime (from C2 runtime) I deleted all the arrays I had and also the LocalStorage plugin.
After that, I changed to C3 runtime and just for sure, rebooted the engine (open - close). Then, I created the arrays again and also the LocalStorage plugin. Then basically re-created everything (mostly copy paste the events & actions from C2 runtime project) and it seemed like it still does not work.
After testing out multiple different things, I'm pretty sure the conclusion is that in my C2 runtime project, I used an action "Insert new element" for my arrays. It didn't work the same way on C3 runtime, since it didn't add totally new elements. After switching to C3 runtime, I had to change the action to "Push new element" to get it working.
After this, it started behaving like it should. So yeah, it seems I'm finally able to continue using C3 runtime!
Unfortunately, this was not any type of "ultimate fix for the problem". More of a "try out a lot of stuff, lose the sense of time and space and end up having stuff working, without knowing why".
Just wanted to type it here, if for some reason, someone else finds it useful. Probably not though.