The good news is, it rained last night, and my basement still isn't leaking.
The bad news is, I finally came across an explanation of why the water problem has to be taken care of outside of the house. Underneath my many layers of patches, I'm pretty sure water is still seeping in through the foundation. Which is probably no big deal, as long as it stays water. Alas, I do not live in a warm part of the country, and eventually this water is going to turn to ice. And expand, thereby enlarging the crack and shifting, ever so slightly, my foundation and my walls. And then melt again, and fill up the now-larger crack, and ... well, you get the idea.
So ... I dunno. As a couple of other people have pointed out, this house has been here for fifty years and it's probably had this same issue the whole time. According to the inspector at the time I bought: "If it were going to fall off the foundation and slide down the hill, it would've by now."
So I'll probably get the gutters cleaned and water the house in the summer, and hope for the best.
Assuming my patches don't leak before then.