This isn't a game like chess. It's not a game at all. Some situations are more difficult than others, based on our past, our spouse's past, our current situation with kids and family, etc. The cheating itself is more or less the same, the behaviors are within very limited ranges.
You deal with people who do wrongdoing and temptation in your job, and cheating is both of that. Nobody gives a shit what junkies say because they are liars, they'll lie to their parents and their spouses and they even lie to themselves. Cheaters are not much different, maybe to a lesser degree than a junkie.
Of course, your situation, your kids, your hangups, your faults, your expectations, hopes and dreams for the future, your belief in who your wife was, which definitely is not true in some respects, maybe are in others, gets some people all jumbled up. It's like when you have 20 tasks to take care of and you can't focus on them properly, can't prioritize them, and keep jumping back and forth. That's what you seem like now.
Don't think about this as a game, think about this as your life. Try to discount your preconceptions of who you thought your wife was, and focus in on what you know for a fact (her name, address, who her parents are - true facts) and forget about what you can't know for sure, e.g., her morals, her dreams, etc. Let go of whatever expectations you had for the future with your wife. She canceled all dreams with you and her. She is canceling them repeatedly every day. Take her out of your dreams and hopes for the future, you can leave a spot in there for her, but don't count on it. That's her decision, and she isn't choosing for it now.
You decide what you want in your life that you and you alone can control. You can tell your wife what you want, you can ask her for it, you can demand it, you can call it whatever you want, but asking her is probably the most accurate. Think about the junkie's words and ignore all her words, pay only to her actions. You know all the lies of the junkie, the ones you are catching, the wrongdoers you come across, and the various lies with words. Wrongdoing is similar across various wrongs done, so cheating and drugging is not all that much different, the lies told are the same. So you ask your wife to do the things you want, and you let her do those things, or not, and gauge it by her actions, not her words.
Asking does not mean you are saying "pretty please with a cherry on top," all polite-like. You can say "I expect this from you" and be a little forceful in your wording, but at the end, it's her decision to do it or not.
I personally don't like ultimatums or threats. Number one, usually the ultimatums are already silently there. For example, "If you don't stop cheating, I will divorce you!" Really, do you think she doesn't already know that? I think all of us understand when you cheat, your spouse might divorce you. And when you get caught, and then keep on cheating, your spouse really knows that. So why say it? It makes you feel better, but I think it makes you look weak. Especially when a week goes by, and she is secretly still cheating, and you're going to marriage counseling in the meantime. Number two, if you don't follow through, you look weak. Giving an ultimatum limits your options for possible future circumstances that you or her change and you can't foresee now.
1-This is not a game - it is your life.
2-Tell her what you want from her.
3-If you feel the need to give an ultimatum, then tell her something like "if you don't accede to my requests, I will do what I feel is necessary, including divorcing you, when I'm good and ready."
4-Start moving forward. Junkies usually don't get their shit together. Cheaters don't either. Cheaters have a better chance to recover in my opinion, but I still think it's at best a 50-50 proposition. My experience, the longer it takes to figure it out, the less likely she ever will. Moving forward might be getting an attorney and starting the process of how will it be without her in your life.
5-If you feel like you should tell other people, your family and friends, about the affair, then do it. I did not tell anyone because my wife didn't and she did everything I asked and I didn't really need it, I think I might have to support them more than the other way around. But if I had divorced, without a doubt I would have told my family and close friends why it was happening.
6-Try to act normal. Don't make up weird rules like "I will leave from 8 am to 5 pm, then I'll be home from 5 pm to 8 pm, then I'll sleep in my friends house or in my car ..." Live your life to the extent you can, don't jack around your kids with stupid shit your wife wants. She is so much messed up in the head than you, it would be like you changing your behavior because a junkie told you that you should. Crazy.
7-Try not to buy into stupid shit from your wife, her ideas, like you need to stay away from her so she can figure out if she misses you. And to go on "dates" with her, like you're supposed to be courting her. That's what she wants, and it's messed up. You two are married, you have kids, you had vows together forever faithful, so if you want to both go out together and have fun together, that's great, and if you want to do things that make her happy and she makes you happy, that's great, but it's not a date and you're not courting, and you're not going to decide "I didn't enjoy the date so I'm not going to see you anymore, I'll go out with other man next week so give me a call in a few weeks and I'll see if I miss you at all." Don't put your marriage on the same level as dating. Don't let her messed-up-ness infect you. Stay normal.