There are several similar planes, most are NAOCs (National Airborne Operations Center), which look exactly like AF1, but without the seal. I remember on 9/11 when every plane in the US was grounded, for several hours I could see a single NAOC circling WPAFB. They are typically reserved for the Chiefs of Staff.
The principal of my kids’ grade school told me this story. The school was located next to a lake that served as a seaplane base. For some reason a pilot, didn’t know about the grounding, forgot, had a brain fart, decided to go for flight. He just got off the water and when an Air Force jet screamed in at tree level and forced him down.
The Principal said all the kids on the playground stopped dead in their tracks in astonishment at the spectacle.