My gut feeling is that he has done a lot more than you know. He wants to like himself so he will believe all he's done, in toto, is actually forgivable. So for you to throw him out and say it's over over the tip of the iceberg is making him feel martyred, because after all, it's just a teensy little thing you're so angry about (in comparison to all you could be angry about...) and he's basically a good bloke, even with the full story, so how can you be so unreasonable as to react so badly to a snippet? He's a good husband and father, all you know about could be written off as a mistake, so why the fuck are you so stubborn over it?
It's batshit insane thinking, and utterly selfish, but if I'm right, then it's denial to protect his own image of himself as a good guy. If you're this angry over the first whiff of what he's done, then he's not a good guy. He can't afford to accept that reality - that he's treated you and his daughters like crap - so he will be impatient with your "overreaction" and feeling truly sorry for himself that you're so angry over what to him (and compared to the truth) is so minor. One thing MN shows is that they only ever confess to the minimum they think they can get away with - so your not digging for the full story, which is the usual reaction, but just being, "that's enough for this to be over" will be a huge shock. You really are acting with way more inner strength than I did or most people can. Your sane and simple reaction - that you know enough to know it's done - is very rare. He won't know how to handle it because just as men who do this have a script, I think those of us on the receiving end often do as well, and remarkably (and wonderfully) you aren't following it. Your response is a lot saner and a lot more lucid than the average desperate need to know what, and why (like it matters, and like he'll ever fully tell you anyway). So in essence his being the one making the choices, in the driving seat, and having the affair is smashed. You aren't behaving like the victim, you're behaving like a partner who was badly let down and isn't tolerating it. In essence, you aren't allowing him to control how this unfolds, and that seems to be making him very confused and angry.
I'm afraid I also agree with others that he is being and was being enabled in this thinking by the men he works with. The initial message was sent from a van when he was with others, no? So they knew. It's also going to bolster his sense that he is being unfairly treated.
I agree you need to email stating all contact is to concern only the girls and the formal arrangements for separating and that no other forms of contact will be tolerated for the time being as you need to adjust to life as a single parent. And I am so sorry, but unless you always used condoms, please do get the STI checks done. Almost all are harmless if caught early enough. I know it's the last thing you'll feel like but as someone who found out the ex was cheating at the GPs when my symptoms turned out to be chlamydia, pre-symptom catching is always best. I had an ectopic pregnancy which is possibly related. Please, you're still so young and may want another child with the lovely man who lies in your future (I know it probably seems impossible now, but truly life can surprise you) so please, be smarter than I was and get yourself checked out.