Consider this datasheet for a family of Zener diodes. All of the parts 3V and under have their leakage current rated as some current at 1V. I've seen this is multiple family datasheets, but it really doesn't make sense. There is no relationship between the test voltage and Zener voltage. I could see a spec such Zener voltage at 1uA having utility. At least the test condition would remain consistent and have the same meaning for each part in the family. Why are Zeners spec'd the way they are?
Leakage current in processes such as these are not primarily controlled for. As long as the devices met some typical value at test then the product is released. The reason for this is that leakage (in it's various forms , Band to band, tunnelling, Hopping) is driven to a first order by impurities and defects which are by definition random.
Principally these devices are not usually operated in any really different mode other than as Zeners, so this is measurement is more of a reality check. And while I can see your argument, if the leakage at 1V is X then it should behave predictably at higher voltages. Of course this is swamped by the zener current once it hits its knee.
Your alternative suggestion, while also understandable would be very hard to do in a reproducible way. Here is a snip from that data sheet.
You can see from the curve in the 3rd quadrant that a fixed voltage (a vertical line) with a variable curve (variable from sample to sample) will show a little variation in the Leakage current. Due to the slope of the curve the sensitivity is low.
Now, flip it around, pick a fixed current as you suggest (a horizontal line) and imagine multiple curves occupying the 3rd quadrant from device to device. Due to the slope of the curve, a slight process variation would manifest itself in a wildly varying voltage. So much so that it would be useless as a measure.
When using a zener diode to protect a sensitive input from overvoltage, you want to know what effects it will have on the signal, which is typically at a low voltage, well below the zener "knee".
Therefore, you'll want to know its low-frequency and DC loading effect (leakage) and its high-frequency loading effect (capacitance).