What is Cut-Off Current in BJT transistors?

when there's 20 volts across collector and base there's no current flow, but when the voltage increases to 30

No. You are reading too much into the numbers. You are treating the numbers as a curve or trend, which they are not. Nowhere does it say no current flows at 20V.

some cut-off happens

Also, this sentence doesn't make sense. "Some cutoff happens" doesn't mean anything. It's like saying "some fast happens" for a car. Maybe you meant to say "some cutoff current flows"?

That is simply one measurement taken under the specified conditions.

Forget the fact it even says cutoff for a second. It does not change the meaning or interpretation of the measurement at all. The cutoff label just tells you this is intended to give you an idea of leakage currents when the BJT is off (in cutoff), but doesn't change what the measurement is saying.

i.e. Disconnect the emitter so Ie = 0, apply 30V between C and B and you get 1uA flowing through C and B. In other words, wire the BJT as just a reverse-biased diode using the collector and base junction and measure the leakage at 30V. Nothing more and nothing less. Maybe that's what you were missing? The emitter was disconnected. That's the only way they can 100% make sure Ie = 0. So the leakage current measured must be between B and C.

When there is 30V across the collector and base (which is the breakdown voltage, so there had better be no more than 30V in any circuit!), and no current following out of the emitter, (for example, if someone's connected a 30V battery and meter across the collector and base), at most 1uA flows between the collector and base.

Ideally there should be none at all (the transistor is off), but at the maximum voltage for the CE junction, a little leaks through, but no more than this even at this extreme voltage.

A similar analysis holds for the emitter-base junction with the other figure.

I assume this is considered a useful measurement because the reader is expected to assume there will be the same or less leakage at lower voltages.

Think of it like this, if you have a water tap it will have a maximum pressure it can handle. That will be in its "datasheet". At exactly that maximum pressure, even if the tap is off, a little may leak through the tap. That's what this figure represents.