You can buy EPDM repair patches and the glue.But it can do harm, because it's bitumen-based, which will attack the rubber because of the solvents.
I appreciate the help, but I can speculate on what will work myself; I'm still just looking for specific advice from someone who knows the best method of first identifying what my roof materials, and secondly, what is the professional way to repair it myself.
Don't use bitumen repair gear or anything else that will foul the surfaces. The repair patches have a peel-off backing.

