After reading three pages I am still clueless. Yes, other people have different tolerances, and I'd certainly never sit in front of someone at the movies or take a parking space someone else was waiting for. I naturally assume the best of everyone. I've taught my children to be polite. We'd likely have figured you were friendly b/c you were lunching in the park instead of a table at a restraurant. Unless my child/pet/self was actually within a few feet of you or doing something to obviously annoy you besides running to a lake/playing in the grass/taking a picture, I'd certainly never guess that our presence in a public outdoor area could possibly bother you that much.
If the couple had looked especially intimate, or been making out, then I'd have headed the other way once I noticed them but really, it's a public park. You don't own it and there's no way the mom could have known you'd be offended or wanted privacy in advance. To assume she did, and was rude, borders on more than a just little self-centered to me.
To resent her presence in "your area" from the moment you heard them begin down the trail is what seems rude to me. Because you didn't want to be bothered with children at that moment you probably noticed them much sooner than they noticed you. I'm sorry your moment was interrupted but frankly, in a public park it could have been a school bus full of kids on a field trip instead of just one small family unit.
This could be a family with children used to going to that spot. It could also be someone who's never been there before and unaware of other options. More likely though, mom & grandmom probably never even noticed you prior to getting there and once there, if it occurred to them then, it sounds as though they'd have had to climb back up the hill to leave and walk further on to choose another place. Maybe there's even a disability that isn't visible to make them not want to go much further.
FTR - I apologize for my children, pets, self - & pretty much anything else you can think of - constantly. My saying "sorry" to someone we pass doesn't mean I know I've intruded or offended them. It's a courtesy - my way of being polite, if you will. In reality it's almost as perfunctory as me saying "hello", "how're you?", or instead of/in addition to "excuse me" in an aisle of the grocery store. Maybe, like me, she was brought up with lots of Catholic guilt and apologizes to everyone for everything, even though there's no visible reason to.