It’s common and desired design to not power the LED while board is not powered by primary source. Would you like to waste your battery with that LED?
You probably would have find answer here, if you didn’t lead the topic to weird voltages and multiple pins… 
Nice that you solved your problem!
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I’m glad that you “found the problem”!
Regarding leds: there re two: one is connected to gpio13, other is “charge” indicator. Since there’s no 5V on the input it could be that led was on when you’ve had 5V present, but it’s not anymore. I do have some esp’s with battery function and led is blue when there’s no battery connected or battery is full; and red when it’s charging. Of course, led is off if i power the board to 3.3V.
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Yep, and even it’s right there on the schematic. The power led is only powered by VBUS which comes from USB. Indeed it makes sense, you wouldn’t want a led on when running on battery.