I have a neighbor who has two kids this age - both toddlers, about 10 months apart (!). She hates being home and is always wandering the neighborhood looking for other parents to talk to and kids who will play with her children and give her a break. I get it - her two are REALLY high maintenance, even judging by the very high bar of toddler maintenance. I bet there is some of this going on - she's planting herself in your yard a) because it's nice and b) because you guys are outside a lot and she wants company and someone to take the load off of entertaining her kids. It's hard but you're going to have to lay down the law by explaining that you guys have such busy lives that you like to have some private family time in your yard and that you'll invite her over when the timing is good.

sheesh, just be direct. (I didn't read all 11 pages, so sorry if you've answered this.)
I'd just say we would like some private family time now, thanks. If they don't get it you'll have to tell them you just don't feel comfortable with people coming in whenever they feel like it. That you are only comfortable with people coming over if they are invited. Say it nicely, but clearly. If the child comes alone, send them home (or take them home, if they aren't leaving.)

A dog...Cujo perhaps.