Meantime, Marian and Sherry were clearly right, with every picture and description I could find being dead-on, and so now I finally know the name of my tree. Yellow Transparent. Thank you!
Sherry told me go pick them, pick them all right now. Turns out they’re supposed to be ripe the first of July and they store better if you pick in the middle of June. I had no idea. And so at dusk I went out there and opened those clamshells; the biggest apple came right off the tree at the slightest touch and the others came inside as well, along with a bunch of plums.
So. I have two apple trees that bloom together but ripen far apart. That’s actually pretty useful. And I imagine if I want pectin in my plum jam I can just chop a Yellow Transparent into the batch, as natural and homegrown as you could ask for. Cool!