How can I create a layer that separates a photo into black, white, and a specified number of grays?

Lots of great ideas here. I'll play around with them and see what works best for my needs.

Thank you all for your help.  I do appreciate this community!

Another way to do it would be to use a Gradient Map adjustment layer, with a Gradient that puts stops "on top" of each other to give hard boundaries. These are super fiddly to set up.

You can see this one has a stop that's not quite right.

Cutout allows up to 8 different colours or shades of grey, but does not force full black and full white.  That would be easy to fix with curves, levels or Camera RAW

As @D Fosse says, Posterize will give you what you specify. Set to 4 levels, you'll get grey values of 0, 85, 170, and 255. It can look as if there are more for the same reason the famous chess board illusion works, or this smoky one from a couple of years ago, but if you open the Info panel and drag the cursor around, you'll see that only those four values exist.

Posterize doesn't use more levels than you specify - but a photograph is usually much more tonally complex in reality than how you perceive it. The brain is very good at abstracting.

To get rid of overly busy areas you could try noise reduction at a very high radius value. If you do this on a duplicate layer, you can mask out areas where you want to maintain the original detail.

Really hard to say without seeing it.