It doesn't matter how you sugar coat it, OP, you basically agreed on a relationship consisting of nothing but sex, thus you were not in a "closed relationship" or a relationship at all, you were fuck buddies, nothing more.
Honestly it just sounds to me like you got the feels but were too much of a p.. cat, to be in a proper relationship, and clearly she wasn't happy with the situation and either wants more 'fuck buddies' or something along those lines.
Now, if you have a written, legally binding contract saying she cannot by law cheat on you, then by all means get your panties in a twist about it, but when she agreed to sleep with you and only you, she was not agreeing to sleep with you and only you, she was just telling you what you wanted to hear, and you were stupid enough to believe it.
She feels no attachment to you, and you, through your reaction to finding out shes cheating or attempting to, shows you do, so just give up and move on, have the balls to be in a real relationship if infidelity is an issue for you, if not and you're only interested in sex, then you've no right, what so ever, to complain about her and any guy she sleeps with other than you, only being interested in sex too, "promise" or not.
Seriously, don't try to paint the picture of her being some sort of whore, because you're just as bad as each other, and calling her your "girlfriend" as in the title, shows just how clueless about your own situation you are, because sex and sex alone does not constitute a relationship, nor does it make her your girlfriend.
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