The demiplane spell creates a door that leads to another plane of existence:
You create a shadowy door on a flat solid surface that you can see within range. The door is large enough to allow Medium creatures to pass through unhindered. When opened, the door leads to a demiplane that appears to be an empty room 30 feet in each dimension, made of wood or stone. […]
It seems that this room is the entirety of this plane. There is nothing beyond the ceiling, floor, or walls. In a sense, the ceiling is the limit of the "sky," although if it has weather is questionable.
Given all this, is the space inside a demiplane considered "outdoors" for the purposes of the control weather spell? Control weather requires that:
[…] You must be outdoors to cast this spell. Moving to a place where you don't have a clear path to the sky ends the spell early.
Best Answer
No, you cannot cast control weather inside a room.
As you've quoted, your demiplane is a room made of wood or stone.
When there are no mechanical in-rules definitions, we generally go with the normal language. In this case, in normal language rooms are indoors, it's what makes them rooms. Being outdoors is not being in a room.
It doesn't matter that the entire demiplane is this room, it is still a room. The demiplane is not an entire world, just an extradimensional space.
Comparison to another extradimensional spell
This is similar to mordenkainen's magificent mansion, which also creates an extradimensional space. Both the room of the demiplane and the mansion from magnificent mansion are indoor structures. They aren't outside so therefore don't pass the gate required for casting control weather.