Those two drivers you linked are the same driver from the same place Kaidomain. LHT flashlight at Ali is KD’s Ali store.
The driver uses a PAM2803 boost IC. It will work with both Alkaline and Li-ion. When the voltage is above the led vf it’s direct drive but the resistance of the components the current flows through plus the feed back resistor reduces the current to a reasonable amount. When the voltage is below the vf of the led the PAM2803 boost the voltage. It does a decent job for its size and cost. Alot of drivers still use this same IC to do both types of batteries.
There’s a lot of info on this forum about using the PAM2803.
http://budgetlightforum.com/search?q_as=pam2803