I flew for a charter company that used a FAA approved callsign. The aircraft number was a coded three letters, plus two letters from the aircraft N number. The callsign was then placed in the remarks section in quotation marks.
On ambulance flights or organ transports, we would sometimes include the nature of the flight in there.
For training flights, you can include comments that it's a training flight, or other pertiment information. Whatever you think is appropriate, really.