Door hinge to living/dining area and kitchen lighting

Correct, that is our storage room.

Precisely because of the storage room, I would hang the door to the kitchen/dining area on the inside right. If someone leaves the door of the storage room open, it will bang, no matter how the door to the hallway would have been hung.

I personally think it is nice in your case if the door is not inside the room but opens outward. (So haben wir es selbst geplant) Although the door to the storage room is already inconvenient. What will go into the storage room under the stairs? Things you need all the time, or rather rarely used items? I would base it on that.

The room is something of a pantry, something of a storage room for vacuum cleaners and the like. It's an addition to the somewhat compact utility room.
We also considered opening it into the hallway, but despite the glass, we thought the risk of it breaking is too high. Especially once there are children around.

I only see a door to a storage room under the stairs - right?

Yes, that's true, I looked over it too quickly.

The room is somewhat a pantry, somewhat a storage room for vacuum cleaners and such. It's a supplement to the somewhat compact utility room.
We also considered opening it into the hallway, but despite the glass, the risk of it breaking is too high for us. Especially once there are children.

Then the chamber will probably be used a bit more often, meaning also when cooking.

Then I would either choose a sliding door (favorite), or open the door towards the living room (alternative, if it can't be done otherwise).
Otherwise, every time you need to get into the kitchen, you'd still have to go around this door.