I found this one really, really intriguing as on first glance, there could be at least three things going on here. I made my mind up and then read the first thread and had my doubts answered. Sorry OP but you are not going to like what I think is happening.
Just based on this OP, OP...
The woman's messages to DP are an attempt at very careful, near safe, levels of flirting but as a PP said and I agree; it's boundary testing. Now why should someone want to do that?
Equally, DP's reaction is off as well. If he genuinely believed that this woman wasn't flirting with him, why show the messages in the first place? If he did believe it then it makes much more sense to show and share with his partner as part of a good relationship. So he's showing flirtatious messages to his partner but saying he doesn't think they are, he thinks they are weird. On this point he is correct, they are weird.
So second question, why did he show OP in the first place because neither explanation makes sense, unless he's doing it for another, yet undiscovered reason.
Finally the lack of reaction from DP. This is very telling in what I think is going on. He's drawing her in. He's baiting the hook much in the same way we all go crazy over text message response times, he is deliberately not answering to keep her messaging him. Cunning fox. And yes, the 'x' is INCREDIBLY telling, from both parties. As other PP's have picked up on.
So, what might have happened? In my opinion, DP is self aware; he knows what he's doing. I'd even go so far as to say that the whole scenario has been deliberated engineered in order to draw narcissistic supply from the OP. And of course the female cyclist too.
You see, the problem with the other female is her sending flirty messages to a partnered man. It's not on. Seeing the actual texts from her show little or no empathic reaction, she just wants sex. And of course she would because she is a narc herself, a somatic.
Unfortunately and I can understand how it might have happened, the female is not aware (usually are not) and because an aware narc is far better at hiding those nasty traits and also mirroring back false empathic ones, she's mistook him for an Empath and thinks she can manipulate him for her own supply. Case of the hunter becoming the hunted. Who is predator and who is prey?
In this case the predator is the DP. The messages are being used to triangulate you with the OW. In that regard, OP, don't worry about it. See it for what it is. I doubt very much indeed if the relationship is physical or indeed might become that in future. But OP, that is not necessarily a good thing in your case because now you are in a bond with an aware narcissist.
I know the above is clinical but honestly, if a poster asks for a hand hold; they get one. If they ask a question then I will answer.
I'm sorry OP 
you DO deserve better than this.