Oh I am sorry OP - it is so hard. I have been through this.
I would suggest unless you have reason to believe she really isn't a safe person to have around them, then I would not ask to meet her. It will just cause you emotional strife and make it harder to separate yourself. If either of your DC have particular needs, you could always send an email to your ex to list the things you want to make sure anyone spending time with them know and ask him to pass it on (if he's useless at those things).
In terms of 'who she is' I think 'friend' can be ok, as it's a language kids get (my dd at 7 wouldn't know what a girlfriend is anyway and didn't understand ideas of fidelity/affairs to question any of it), but seeing them being physically affectionate may trigger confusion for them in a way that words might not. So if presenting her as a friend means he holds back on that side, that is probably good. I would use something like "special friend" to recognize that there is something different from other friends, without sexualizing it unnecessarily...?
But in the end, these things are not in your control. If you do have a collaborative relationship with your ex, (and him telling you his intentions in advance suggests that you do) then you could ask him for your preferences. I asked my ex to keep visible physical intimacy to a minimum to start with, to not leave them alone, and introduce her on a day when dd was coming back to stay with me, so I could be there for her if she was upset or confused (she wasn't). I also gave dd a little necklace and dressed her in something I had specially got for her, which to my imagination (not expressed to dd of course), was a talisman to be like a forcefield of my mama bear protection/ motherliness! Then I lined up a careful few hours for myself. Phone calls with friends and a scented bath to spoil myself. It was very painful but better once it was over.
In the end, the OW- despite her obvious character flaws is wonderful with my dd, by any objective measure- and painful though that is in one way, for dd it is way better than being with someone unpleasant or who doesn't care about her.
I hope it goes ok for your dc and for you, OP. 