First of all Annie, I can empathise with your enormous shock. In one fell swoop, your H has fallen from the pedestal he was once on and there must be moments now when you look at him and wonder whether you knew him at all. It's a terrible loss of innocence.
Unfortunately however, what your post tells me is that your H is trying to minimise what happened. His words are very revealing. This was not a "silly game that had got out of hand". He knew all along that this was not a game and it was not silly. A fantasy perhaps - and a bit of escapism, but he would have known that this wasn't a game.
You also say that he "hugely regrets it if he gave the wrong messages by receiving and replying to her texts(she initiated them)" Well, if you're texting a woman in sexually graphic terms, she can hardly receive the wrong message, can she? The message is pretty clear - he wanted sex with her. Also, while I have no doubt that she may have initiated the contact, he appears to be minimising his own culpability by pointing out that she did so. He could have said no.
I'm afraid that since they were also meeting regularly, I cannot imagine that they discussed the weather and world politics on these walks. It would be stretching credibility too far to imagine that having built eachother up into a sexual frenzy via text the night before, or earlier that day - they didn't do anything other than walking the dogs.
It doesn't surprise me that OW is confirming his story, because no doubt they are sticking to the same story with their spouses and presenting a united front. It suits her to verify his story, because that's what she's telling her H too.
Therefore I endorse the suggestion that you meet this man and swap stories. I would bet you'd find some discrepancies in their accounts then.
I would be inclined to agree that nothing physical had happened, were it not for the regularity of their meetings. I would also believe more readily that this would not have progressed to a physical affair if they hadn't been rumbled by her DH. Neither of them can claim that they had realised the error of their ways and had decided to put a stop to it. This would have continued if they hadn't been found out.
This doesn't mean your H is a bad man. He's someone who engaged in a friendship that started off as life enhancing and rewarding, but then turned into something else entirely. It is very common, especially amongst good kind people who think that they are immune to an unsafe friendship, because their marriages are happy. These very people are the most vulnerable, because they don't see the threat until they have become hopelessly addicted and infatuated.
You can overcome this and have a much stronger and even happier marriage, but in order to do this, you need some real honesty.
In the wake of any affair, emotional or physical, the betrayer must be honest about the indisputable facts - what physical contact took place, when and where and who knew about it. This is a must.
Unfortunately, what takes longer to establish is motive and reason. Often, the betraying spouse is lying to both you and most of all himself in the initial aftermath. The lies to you are all about minimising and protection, of himself and you. He will be telling himself that he is lying so as not to hurt, but what he won't acknowledge is how much he is protecting himself by these lies too.
It is essential that you both get some answers about why this happened. But first, he has to understand himself and why he departed from his values - why he gave himself permission to get involved. Often, this will be brought out by some counselling on his own. I would insist on it.
Not Just Friends by Dr. Shirley Glass will really help you both to understand what has happened here.
I hope that your H will give you more honesty, because I think that's the only way you can move on from this. If you've constantly got nagging suspicions, you'll never really be able to heal - and you will feel an additional sense of grievance that you were lied to even after discovery.
Good luck and keep posting.