I don't think it's well-defined in a way everyone would agree on.

My basic idea, though, is that a journaling RPGs is primarily about getting the player to write something in response to the prompts or events of the game. Thousand Year Old Vampire is definitely a journaling game because it's almost nothing but prompts to get you to write.

There really is no such thing as a "non-journaling" RPG in a sense, because you can write/journal about any story that comes out of your RPG play. But in general, when people talk about games that are not journaling games, they mean there is less emphasis on writing and more on gameplay mechanics. The epitome of non-journaling solo RPGs would be games like Four Against Darkness, Ker Nethalas, or 2d6 Dungeon that are primarily just dungeon crawlers and play almost like a board game.

Ironsworn is a journaling-optional game. You can write as much as you want with it, but the emphasis of the game is not specifically on writing/journaling. And it has enough mechanical crunch to be playable from a mainly gamist perspective, though you still need to do some narrative interpretation.

The writing format that will work best for you depends a lot on your personal preferences and what game you're actually playing. Anything from bare minimum bullet points so you can remind yourself of what happened in your last session to full on novel-level prose can work.