MacBook Pro 2009 doesn't boot without charger

I have bought a used MacBook Pro 2009 from eBay.

I have charged the battery to 100% (worked fine), and booted it. Everything worked, except that I can't boot when on battery. The battery itself is fine, so I can connect the charger, press the start button, and it starts. I can then disconnect the charger, and it will keep running. But without the charger, it will just not recognize the power button press.

I tried resetting the SMC (Shift+Ctrl+Option+Power Button), and resetting the PRAM (Cmd+Option+P+R like 5 times), but nothing changed.

Any ideas?

Update: Now it won't turn on even with the battery inserted, and it also does only charge if I try several times to connect the magsafe adapter

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.