I took my American Standard Stem in to the Home Depot. They compared to to others and said that it was a 2C-14C. Then they gave me replacement Nu-Seal washers and said that there is no seat for this. Is that possible? Here it is with the new washer attached.

That looks like a compression type stem to me. I would expect a seat. Shine flashlight into stem hole. You should be able to see whether there is a seat.