Then it is working correctly?
As I said, I suspect some of the circuitry is probably powered by the output, which means without any input, the LED will draw its power from the battery connections.
Adding a schottky to the output of the module will ruin the voltage regulation, since its feedback is taking before the diode. This means the output voltage will be a function of current, as well as temperature (diodes forward voltage is dependent on these two) which makes it somewhat dangerous for charging lithium batteries. I would leave the diode out, as I doubt it draws that much current if left connected to the battery.
I would trace out a schematic from the module, shouldn't be too difficult, and you can see how it works, and possibly modify the board to obtain the functionality you required.
Also, just because a website claims it is specifically for battery charging, doesn't always mean that is the case. Sure, most websites are fairly accurate in their description, but from ebay/aliexpress/amazon often descriptions are just copied/pasted from elsewhere by those who dont' know much about electronics. It certainly looks like that module uses one opamp for constant current, and one for current cut-off (fully charged) but without tracing out a schem, I dont' know.
I believe I have a found a schem, which does indeed show the low current cut-off indicator:
https://www.aliexpress.com/item/DC-DC-LM2596-Step-down-Adjustable-Power-Supply-Module-CC-CV-LED-Driver/32667762137.htmlIf you follow the paths to the LED's I believe the rightmost LED could possibly be powered by the output, but its hard to tell.