Your light looks like a typical MagicShine light with optional head mount. (MJ-808E) If it doesn't say MagicShine on the light itself, then this could be either the same light made by the same OEM manufacturer but sold through a different channel as a no-name light. Or it could be a knock-off of the MagicShine light.

The battery does not need to be discharged before charging. Green light on the charger means that the battery is fully charged. And there is no danger in leaving the charger plugged in with the light green. Of course, all that assumes that the charger is implemented properly, which no one can guarantee with a no-name light.

Basically, with a no-name product sold without support and/or instructions there's always a notable "use at your own risk" element.