stairs
I'm going to rebuild a 6-step stairway in my house.
Doing stringers calculations right now – but I'm not 100% sure if I supposed to use a layer of OSB on top of stringers prior to finished treads?
Like in this video here I'm seeing that stairs have a layer of OSB.
But in this video here guy seems to install finished treads right onto stringers.
Which is correct?
There are 2 ways to do this, the first way is in essence just cutting the wall at the top of the skirt board and capping it off and setting the handrail on top of the cap that is finishing off the cut wall, sounds a bit crude, but essentially that describes what you will have.
The up side to this is you will not need to do a floor repair, depending on where your wall ends where the newel will be.
The other way to do it is by removing the wall entirely, repairing the floor, removing the treads and risers, and set longer ones with return nosings on either side. and re-support the stairs with a new walls under the existing carriages.
This is the most work, but will give the best result in my opinion.
Jan. 2, 2014 edit
Here is the work I did in 1989 that shows what I did.
This one shows the best example of the effect I was trying to explain. I do not have a shot that shows how the handrail goes past the ceiling, but with a little imagination you could picture what the rail would have to do if the stairs were much wider at the bottom.
Edit #2
I did a little calculating to help, if it may
I think the least invasive way to handle this is to add two more stingers, one right against the existing stairs(essentially just a nailer) and one out at the ultimate width of the new treads. This way you don't have to remove the old sub-treads/risers or change the height of your treads, which would cause you grief when you get to the top of your staircase. Once you've got your new stringers in just apply sub-treads and riser to match the existing construction and you're good to go. The bigger problem is that no matter how you construct it, the landing or the the lower staircase will have to be modified to make the intersection work properly. 
This illustration shows how the problem could be fixed by lengthening the landing and pushing the lower stairs out (assuming you can do this). You could accomplish the same effect by pushing the landing to the right but it looks like your landing is captured on that side in the pictures. Although this fixes the stairs the railing system will still have an unusual transition but it's not too bad depending on how far you need to extend your treads.
Best Answer
OSB is used for one of two reasons in my area:
just builder's grade prepackaged stairs. I can order a set of stairs for a couple hundred and have them installed in a basement in 2 hours by getting OSB (I never do this but could).
in a cheaply built home these could be used as is and carpeted.
I would personally never leave the OSB under the treads unless it would seriously mess something else up. The caveat is if you are using MDF treads, might as well leave the OSB. OSB is just crap and will certainly not take spills and moisture like a solid piece of wood.
Also I would worry that the "give" of the OSB would lead to squeaky stairs. For instance if I left it and used MDF I would just glue down the MDF everywhere on top. If using solid wood I would want to screw them into the risers. I wouldn't want the OSB sandwiched in the middle since I wouldn't trust this to be stable and even.