I knew they had to be standard wired.
There are two standards existing for trailers, 4 wire and 5 wire. The 4 wire system uses a common red bulb for brake and turn. That is one wire for each left and right, one for tail, and ground. The 5 wire uses separate amber turn signal bulbs so it has right and left turn wires, a brake light wire that goes both left and right, a tail light wire, and ground.
The Spyder on board configuration is 5 wire with a trailer wiring converter, since about 2012, to go from 5 to 4 for the trailer connection. If the Bushtec trailer has amber turn signals then it is wired as a 5 wire system. Both 4 and 5 wire trailers exist, I think even new ones today. The Can Am trailer is 4 wire. Goldwing has been, and maybe still is, 5 wire. Trailer harnesses for Goldwing have mostly been 5 wire and the norm for trailers for Goldwings were/are 5 wire. I don't know what the norm is for H-D.
If you buy a 5 wire trailer you need to either convert the lights to a 4 wire system, which does mean modifying the trailer wiring, or add in a converter to go from a 4 wire bike connector to the 5 wire trailer connector, or change the trailer harness on the Spyder to provide a 5 wire trailer connector.
If you buy a trailer with a 4 flat connector one trick is to cut the flat 4 in two and tape back together to fit the Can Am 4 square connector.
In short, in contrast to automotive trailer configurations that you are accustomed to, there is no single standard for motorcycle trailer wiring.