I frequently start and end flights in the air with various aircraft, including the ones you mentioned, to test specific issues, and I’ve never had ATC call me anything but my custom call sign/N-number, nor had my pilot use anything but the custom name I’ve set up. Well not out loud anyway…they might be saying it under their breath where I can’t hear! 
But I’ve never seen this issue while conducting numerous flights I’ve started in the air.
Regards