Okay, so, an update on the situation.
Unfortunately, the issue came up again.. It crashed mid-game and the exact same things keep repeating. Could there be something with TS4 causing this issue to repeat itself, or is it related to my laptop and the graphic drivers somehow just not being compatible with the game/way I play? Or is there maybe something that could be causing for my graphic card to crash repeatedly? I was playing the game with mods, tried launching without and nothing changed. I have the game updated to the most recent patch, so that couldn't be the issue.
Is there anything I can do, except for reinstalling the drivers again? I know this isn't directly related to TS4, but do you think that contacting HP directly could help fix the issue (so that it wouldn't keep repeating itself at least)? It seems that the Intel card works again, but I really wouldn't want to put too much weight on it, as I like to play with the highest possible settings my laptop just can run...
Here's my new dxdiag file attached as well.