Don't invent statements. Second time now you claim 40% variation without backing it with facts.

When your car doesn't start, you may see a super-duper low voltage. Leaving the car alone without load will get the battery back to about 10V. You can short the poles and get it down to 0V, but that is irrelevant. The relevant thing is that the amount of energy is very small when you reach 10V. You will also be able to notice that the inner resistance will be high so when a lead acid battery have reached this level, even small loads will drop the pole voltage.

But once more - note that the article did not specify a battery type. I decided to talk about a lead acid battery. With 10 cells of NiCD or NiMH you have about 2V from full to empty. But more than 90% of the discharge will lie within a voltage variation of about 1V - or about 10% variation.

Look again at the design. The other LED is intended to have 10mA at fully charged battery (but the editing process managed to get wrong R2 value) and go to zero at low battery level. If it "just" changed 10mA -> 1mA, the load regulation of the "ref" LED would be 100 times better than the load regulation of the second LED. With that in mind, it really is quite ok to talk about constant current.

The voltage you get in the wall socket is considered "constant". But it varies more than 10% depending on where you live. And quite a number of people lives in regions where it varies with more than 10% depending of time of day - or which day of the week. If using incandescent lamps, the light from our normal lamp bulbs will vary way more than that LED does - and we do consider a 40W lamp to have a "constant" intensity.

The term "constant" is not an absolute term for the simple reason that no electronics can exist that manages 0.000000... variation. This is a reason why we always have to qualify statements - telling exaclty "how exact" we require things to be.

I mentioned it before, but now I make a direct question to you. Would you be able to view a single LED all on it's own and be able to tell if the intensity is 10% off without seeing the intensity change or having another light source as reference? Because that is the requirements (or even better than the requirement) for "constant" for this application.

Any professional should know that "constant" may sound like an absolute term but almost never is. We get the rainbow because the light doesn't have a constant speed - it's affected by the media...