EDF, you did the right thing, her telling you she won't stop seeng him and won't be completely transparent are deal breakers, you will resent those things and it will not work. Also, the email telling him just friends is totally useless, cause she can tell him whatever she wants next time they hang out.
In my case, OM works for fire station and my W agreed to no communication with him but refused to quit fire station... I resent her not quitting, but I can tell you if I ever find out they are chatting, I'm done. No ifs, ands or buts, if she talks with him on a personal level I'm done.
Keep doing what you are doing, if she wants you she will agree to cease contact and allow you whatever you need to verify.. If not, your setting yourself up to repeat this process.
M - 9 1/2 years 5/5/16 - Bomb drop - 3 week EA 10/31/16 - We sold house 01/10/18 - D Finalized