Bill, welcome to the forum.
Many new owners don't insert their seats forcefully enough into the holes. First try pounding them in with the heel of your hand, a block of wood or a balpeen hammer. If that works, consider leaving your seats permanently installed. Notice that you can collapse them forward and use the existing bungees to secure the backs down for transportation so they don't accidentally flip up (see picture):
If they still won't stay, wrap a little Duct tape around the pegs and try again. If for any reason this doesn't work, just epoxy the darned things in. If you need to remove the seats, you can unscrew them from their pegs if necessary. The screws are accessible by lifting the Velcro over the pegs.
The early Hobie seats were riveted on, so they were originally a permanent part of the boat. Even with the newer models, I always leave the seats attached unless they need to be removed for some specific reason. Also, with the seat permanently installed, that's one less thing to forget! An exception to this might be if you store your boat outside and want to protect the seats from the weather.
PS Looking forward to seeing your strap pictures.