Yup, you're right it must be somewhere between a satin and an eggshell. I went to Home Depot and I bought a gallon of Behr Premium Plus Ultra Interior - Ultra Pure White - 7750 - Satin paint
, 678885070298. It cost $35. I should have bought it from their website because it looks like they sell the same thing for $31. I put a coat on the ceiling and it looks nice. The sheen is a little greater than the primer's sheen.
The paint was thinner than the Zinsser primer. I thought since this Behr paint was also a primer that it would be about the same. Maybe I'm just too used to the Zinsser stuff? The paint job came out good but under scrutiny there are two or three roller lines I can barely see but want to take care of, so I'm doing a second coat tomorrow. I thought it was thin stuff but I don't have much experience with paint.
For anyone who is going to use satin on a ceiling, bear in mind I put a lot of work in before I primed making sure the entire ceiling could be made smooth after using joint compound fanned out to hide the seams, bumps, dimples and imperfections. Then I sanded everything nice and smooth. The entire compound/sand/compound/sand process took a few hours a day for several days.