Heck, I still can't figure out how the receiver (with no antenna input and connected to the TV only with YPbPR1 input) could display a message on the TV when I'm watching the TV Air input
The TV station's DISH receiver is putting out the message.
Here are the steps:
1) The main TV station produces and transmits a broadcast signal from their main facility
2) DISH receives that signal from the main facility and puts it on their satellite service for their customers
3) The repeater TV station uses a DISH satellite receiver to pick up the satellite service and tune to their feed
4) The repeater TV station transmits the signal received from DISH over their broadcast transmitter
The account that the TV station is using to receive their signal needs renewal, so DISH is putting an occasional nag message on the screen. Viewers in your error see whatever that DISH receiver puts out ... including the nag message or any errors. (If the satellite signal was lost due to a severe storm you would see the DISH searching for signal messages.)
It is cheaper for the station to pick up the signal via DISH than build their own connection between the two stations. Or perhaps they use the DISH feed as a backup when the normal reception is down. In any case, you're seeing your local repeater station's DISH receiver.