ppeterlin: My guess is that in addition to maximal discharge current, power banks also have some minimal discharge current, and shut off if the discharge current is below this threshold. They treated my Arduino as "leakage" and shut off.
Yep. That's exactly what the manual says - though not in so many words.
Well, the "power bank" contains a 3.6V lithium battery, so to put out regulated 5V, it contains a power up-converter. The power converter is not perfectly efficient so it uses some power to operate even if there is no load, and of course, you are running the indicator LEDs. To prevent the power converter itself and the LEDs from discharging the battery, it does not turn on at all until you press the "Power" button, and if it detects that it is not being used significantly for 5 seconds, it turns itself off to prevent discharge.