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I have struggled to find an established name for this value. I thought it would be something called the "Schwarzschild density", but no luck. Famously, the [someone's name??]-density of the observable universe just happens to be very close to the density that would tip the universe over into being a very large black hole

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Wouldn't it just be the inverse of the Schwarzschild radius? https://en.wikipedia.org/wiki/Schwarzschild_radius

But given that the universe is expanding, if we're close to this density doesn't that mean that the density used to be higher than the threshold at some point?

This is why when talking about the big bang physicists talk about the inflation field at the start of it. The primitives in the pre microsecond universe have to be different from a black hole otherwise there's no way to un black hole yourself.

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