Yes you are correct. I lap the bore to 1200 grit and the actual bore size stops at .365" because the memory of the bullets bring them back to .366. Not in all cases, but it does work that way many times. PPU bullets are actually made at 365" and they shoot very well, so I figure I am safe if they come out a bit small, but I doubt they will.
The .375" portion of the die doesn't even need to be smooth as long as it's concentric. A good sharp new bit is all that is needed there. Only the sizing portion of the die bore needs to be smooth and polished.
And yes, going down the full .009" in one pass is not a problem. As long as the lead between the .375 and the .366 parts is good, and as long as the .366 part is smooth they go right through. I have done this kind of thing many times in the last 40 years and I am still using the same Rock Chucker to do it with that I was loading on in 1968