a lot of the low-voltage drivers will do that, with ONE li-ion. i assume your talking about 1 18650.
why
well as you see in the first there, it went over, so it really did not perfectally regulate, chances are good it was out of regulation when the voltage of the battery was high. AKA, what correctally driven LED do you know that puts out more than 250 lumens
out of a light.
this is exacatlly how many of the 1.2-4v drivers act, and it is also dependant on the LEDs Voltage or VF thing. it is a boost driver being used for a voltage that (really) gets to high, although it usually functions ok, it often overdrives the led in the first 10 minutes or about 5-10%.
the overdrive occurs when the "boost" driver isnt in a position to boost anything
because the battery voltage is higher than the LED voltage. so the effect is greater or smaller depending on the max charge on the battery, and the led itself your driving.
it is likly to also be a PWM type of levels driver, because on low the PWM pulses were still in overdrive,and so therefore you still have higher spikes When the voltage of the battery was high.
so there are MANY drivers that will work like that.
they range from about 350ma to about 850ma
and can have various ways to have levels TOO.
so what is the drive current you desire
do you want levels and strobe
do you want a few levels or 20
there also exists "wizard" or "flex" type drivers that both buck and boost, are more expencive, usually dont have levels, and will fully regulate from beginning to end, but they are not as popular.
the actual light you mention "
1.16A invariable current to the LED"
has a great heat removal capability, for a reason