The Bodhi is Trevor's car, Franklin's is something else that I think also starts with B but it's that long since I have used him, I can't remember what it is. Buffalo, I had to find it or it would have driven me crazy.
But my experiment was to determine if this was something that affected every character, if another vehicle is saved in their main car's spot and saved with the phone. Saving in bed is like a save-and-reset that puts the world back to normal. Save with the phone, is like a store-your-world kind of thing, so they do work slightly differently. I suspect if you went into Franklin's house and got in bed, things would go back to normal and the addon would get removed. LOL, scratch that idea... I just tried exactly that before posting this and now both have gone.
Edit: After driving around for a while, Franklin's car came back and the other car remained gone.
That house has a fairly unique setup, where there are several vehicles and a garage in close proximity. Trevor has his main parking spot on the front, a quad-bike round the side and then his garage at the back.
I think you can store a motorbike inside Franklin's garage along with a car but again. I can't remember for certain. Edit: Yeah, you can... I still seem to have one in there just inside the door on the left side.