Consider how easily you can change the light bulbs.

I need pictures and you need to show us whaere the electrical is in the staircase . Post all of that here in comments DO NOT start another post

Sconces or pendants on chains are my first thought, or step lights low in the walls or in the skirt board. Hard to know without pictures.

My son in law placed very discreet rechargable motion sensor lights on his finished basement carpeted stairwell walls . They are low on the wall . They are incredible. perfect lighting and extremely reliable and come in lots of designs . If i were you I would start with this inexpensive easy solution and then you have the time to research a more permanent one while providing and attractive safe stairwell.

Currently I have all the plaster removed and am reorienting the bottom half of the staircase because I added a guest bath under the stairs using a Gerberit tank and Toto wall unit. Haven't decided whether to remove the carpet or replace with a runner. Was thinking I would use 2" puck lights, but they would face the open staircase side. Ceiling height at top of stairs and landing is 9' and 10' 6" at stairs bottom floor.

He placed them down low on the wall and they are more than bright enough to light the whole step and surrounding area as one is going up/ down the stairs. you could always buy a couple and try them out. very inexpensive solution.

You could buy a couple and try them out. they give out more than enough light for the step and surrounding area. Very discreet

Why not wall sconces?

These are mine and they light the staircase very well at night.