Rose, is the "pursuit and distance dynamic" part of detaching or something different?
In a nutshell, the more you pursue then the more pressure she feels and the more she will distance herself from you. You break the cycle by pulling back and not pursuing. Typically when you do that the other party will quit distancing, and will eventually start reaching out and doing their own temperature checking/ pursuing.
Even though you are inviting her to what you consider family activities, it looks to her like you're trying to get her to do things with you (which I think if you're honest, you probably are). That is pursuit. Like Rose said, she's free to spend as much time with S as she wants, she doesn't need invitations from you. And clearly from your snooping you've discovered your invites are not welcomed. So putting it all together you want to (drumroll) STOP PURSUING
