MacBook Pro 2009 doesn't boot without charger
I have finally found a fix:

I left it overnight on the charger, sitting at a green light (100% charge). Apprently, that does a battery recalibration. Then, I was able to boot it from battery using the bootup pads on the mainboard. After then also cleaning of the keyboard connector (only like 3 keys were working) with isopropyl alcohol and some scrubbing, it finally worked.