Arn,
I am sorry about the crisis that your wife has just dumped in your lap. It is a lot to cope with, but I am hopeful that if you take what you need from this forum, and leave what you don't, it will help you figure out what it is that YOU want, and that YOU need to help you get yourself out of infidelity.
I think enough posters have now proven that being a stay-at-home-Dad is not a guarantee of being cheated on, because there are plenty of threads in these forums from soldiers who have been in active combat while their wives have been serial cheaters. Cheating does not happen because of the betrayed spouse; it happens because something is broken in the people who cheat.
And while we're at it, it is now public knowledge that the famous actor Colin Firth has been cheated on by his wife. He's famous, wealthy, talented, lusted after by tons of women, and he got served the same shit sandwich as the rest of us. The reason being that cheats don't give a damn about their spouses; their activities are all about them, not us.
The only reason betrayed spouses think it is about them is because of the excuse-making and blame-shifting that cheats engage in to try and justify their actions or portray themselves as a victim. It is bullshit; do not engage with it. Cheaters cheat because they want to, with no regard for the pain it causes, just as burglars steal for the easy money and the adrenaline rush, with no concern about the damage it causes. Their pleasure comes at the price of someone else's pain, not their own.
What your wife did is a classic ploy that has been played out countless times in the Just Found Out forum. She deliberately and knowingly put herself in harm's way, by engineering a situation where there was a ton of potential for infidelity to occur, as long as the man that she had been firing up and flirting with for weeks took the lead, thereby allowing her to say, "I didn't plan for it to happen. I would never do something awful like that...But somehow...It just happened...I don't know how". As if anyone over the age of eight cannot see that the supposed innocent victim acted exactly like a straight man in a comedy duo setting the funnyman up to deliver a punchline. That is why the excuses that are made afterwards are so weak. "I just got swept along in the moment"..."I didn't realise what was happening until it was too late"..."I thought he was just being friendly and giving me a cuddle".
The bottom line is, if she was not prepared to have sex when she went to meet him, she would not have had sex with him. She knew it was a possibility, and she set waited for the funnyman to deliver the punchline after she had set it up.
The fact that she went out of her way to tell you that she supposedly fell out with a different man because he was interested in sex seems like a smokescreen that she used to disguise her chats with these men as something non-sexual, when they appear to have been anything but. If she found one man's interest in having sex with her offensive, why was she so open to it with the other man? Was it seeing pictures of them in social media that made the difference?
They ended things straight away, removed each other from their social media. However she doesn't want to give up her job, which means they will still be forced to speak. She insists she will be keeping it strictly business and not chatting or having 'banter'.
Arn, they won't be 'forced' to speak. They will continue to speak because your wife wants to stay there talking to him. Why isn't she uncomfortable with that situation? If she isn't, she needs to explain why.
Given what she has just done to you, she should be offering to leave the job if you need her to, not dictating the terms and telling you she is going to stay there and act in a 'safe' way in future. How will you ever have any idea what she says to these men, particularly if she has already established a reputation as a woman who likes dirty talk? And what happens if the other man tells his friends about his easy score? Is that going to put them off trying it on with her, or encourage it?
She has proven that she cannot be trusted in that environment, and she needs to be working 110% to prove that she can be a safe and trustworthy life partner for you, not making efforts to remain in the situation that led to her arranging a hook-up for herself while lying to you.
What is worrying is that she is either doing that knowingly, or she genuinely cannot see how bad her boundaries are, and how open she is for exactly the same thing to happen again. If she could be trusted to prevent this from happening, it would never have happened once. She needs to be out of that place, and away from a conveyor belt of men who are never going to stop flirting or being outright sexual with her in their conversations. Clearly, it's what those guys do at work, and if your wife enjoys it as much as you say she does, how long is she not going to get drawn back in by it?
As far as your decisions go, it is within your rights to take as long as you want to think things through, particularly what you need her to do (and not do) for your mental and emotional well-being if you are going to stay. It sounds like she is trying to rush you into a decision, and that she wants to minimise what has happened as quickly as possible. As tempting as it might be to try and bury this and pretend it never happened, that is not in your best interest, or hers. If she does not fix and change what is broken in her and her values, she is likely to cheat again, because nothing in her, the way she is at the moment, prevented her from cheating.
There are all kinds of steps that you can take, and things that she can do to engineer a change in herself, and plenty of people here have real-life experience of what works and what doesn't. I hope that this forum will become a powerful resource for you.