I would run a rail light down the stairwell or convert that to a can light and add one more.

For the room, I see its kind of a hang-out kids area, but what is the style you like?

What is a rail light? Like movie theater lights to light up the way? For the loft, it’s an office/play room area. Very casual relaxing and not for public entertainment

its basically sleek track lighting and would let you light up that stair well and maybe even hang some pictures on the wall.

for the ceiling get a semi-flush, but not too low, that will let light out of the top and bottom, the white ceiling will help spread the light around.

https://www.homedepot.com/p/Minka-Lavery-Harbour-Point-3-Light-Liberty-Gold-Semi-Flush-Mount-Light-4177-249/205942878

Wow! I never thought about that for the hallway. Love the light for the loft just will try and find that style or size in brushed nickel

One trick you can use right now is to turn the slats of the blinds so that the light reflects off the ceiling.

If you own the home, hire an electrician (or get owners permission) and have recessed lighting installed. You’ll never regret it. Ample light; problem solved.

Recess lighting where in the stairways or loft or both

+if you go recessed put 4 in the loft one at top of stairs one mid-way..

What kind of lighting is this called?

Aimee, recessed lighting both in the stairwell and the room. And to answer your question, that is a type of cove lighting