Walls – What do you call the clips for attaching a dado rail

While Chris Cudmore is right about the real fix, you could work out a temporary fix as follows:

Brace the door

• Get two screws per hinge that are about the same thickness as the ones shown in the picture and about 1/4 shorther than the thickness of the door.
• drill a pilot hole on the inside face of the door toward the outside spaced midway between the current screw holes. (Note that these are not on the same face of the door as the hinge screws). The hole should be thinner than the screws selected and stop about halfway through the door.
• force some carpenters glue or white glue in the cracks at the edge of the door that is under the hinges. (you can force it in all the cracks if you have several clamps).
• clamp the door to pull the crack together.
• set the screws in the pilot holes.

Rebuild the hinge holes

• lightly coat dowels or other thin wooden rods (you can use chopsticks) with glue
• force those rods into the screw holes for the hinges. Add slivers of wood to fill any gaps.
• after the glues have dried (next day), remove the clamps and trim the dowels flush with the recessed hinge area.

Reset the hinge screws

• drill new pilot holes for the hinge screws
• screw the hinge screws into the new pilot holes.

Quite a bit of fuss, but the only real cost is a few decent clamps. You need screw types, not spring clamps.

Or you could ask your landlord to fix /replace the door.

Luckily your mount has slots for the bolts so you have a bit of room to work. Get another (stronger, higher quality) lag bolt and put it in right beside the broken one.

Leave a tiny bit of room between them if you can, but make sure it still solidly hits the stud.

Predrill the hole. To find the right size, hold the bolt up behind your drill bit. The drill bit should cover the entire shaft, but not the threads. Make sure you drill to the same depth the bolt will go to.

Rub the bolt threads on a candle or bar of soap to lube it up a little if you think it is going to be hard to drive in again.

Drive the bolt in slowly. Don't use a drill unless it has a low torque screwdriver setting. You might want to tighten it with hand tools anyways so you don't risk snapping this one too. There probably isn't room for a third try!