My wife recently asked me if our relationship could be open, so that she could date other men. I agreed to it only if she was completely open and honest about everything that happened between her and someone else. No secrets, no lying, 100% honesty about everything. For example, if dating got more serious and sex became a possibility, I needed to know about it beforehand.

When my wife met someone, she told me repeatedly that the other man was just a friend, more like a brother, and that she wasn't interested in sex with him at all. She ended up sleeping with him and lying to me about it. I also found out that she slept with him without using any protection with full knowledge that he had STDs...then slept with me afterwards without protection several times (it was months before I found out the truth and by that time, I was already very much exposed).

The trick is this: a year or two prior, my wife asked if I wanted to sleep with other women, because a friend of mine (a woman) really liked me. Initially, I told her no, because it sounded like a trick. Eventually, I did sleep with another woman, but I told my wife before it happened that it was a possibility and then afterward that I had done it. She was made completely aware and I was completely honest about what happened. There were absolutely no secrets. I also used protection while sleeping with the other woman.

My question is this:

Since I agreed to allow my wife to date someone else, but never agreed to her sleeping with another man without first informing me, and then was lied to repeatedly about it happening, AND STDs were involved, does this constitute infidelity / cheating? Since I was given similar freedoms prior, but handled my situation very differently, does it invalidate what she did by lying to me? How does this affect what I might be on the hook for financially since she is completely dependent on me? We have a child, but I am also committed to minimizing, if not completely eliminating, any potential negative impact this will have on them.

If anything I have said needs more details, do not hesitate to ask.

EDIT: This takes place in Alabama, USA.