The tread depth (distance between 2 risers) is 9".

OK, so if the "run" is the horizontal distance between risers, then
the actual treadth width is run + nosing. Not sure whether "tread
depth" refers to run or actual tread width, I think it usually means
run.

Anyway, you have a 8" rise 9" run stairs now, and as long as you use a
finish material of uniform thickness, you'll have a 8" rise 9" run
stairs when you are done.

Each rough tread has a 1 1/8" nose. The finished laminate nose
pieces only have 1/4" or so noses.

If you think about it, I believe you'll find that "new nosing = old
nosing + laminate overhang - laminate thickness". The "- laminate
thickness" part is based on using a finish laminate riser.

BTW, the 1997 UBC requires a nosing between 3/4" and 1 1/4" on stairs
with solid risers.

Cheers, Wayne