Power supply or battery problem after damage

Hi I have a MacBook Air A1466 (early 2014) and after water damage I have the following problem:

The computer does not start when I press the power button, but if I have not connected the charging cable and unplug the battery, then connect the charging cable again, the computer starts by itself. Then everything is fine and working well until I shut down the computer. With a restart, the problem is not there.

The Power button is not damaged. When I am at the login window, the charging cable is plugged in and I shut down the computer, I can normally start it with the power button. But only with the charger cable plugged in!!!

When I am logged into my account and then shut down the computer, it will not start when I press the power button, whether or not the charge cable is plugged in.

Then again I must unplug the battery and connect the charge cable.

I already cleaned all the parts with Alcohol.

Maybe someone has had the same problem and can help me.

Is it a motherboard problem or the small part where the charging cable is connected to is damaged?

I also tried to reset the SMC, which seems not to help.

Thank you so much for reading and excuse my English.

Nobody have an idea? I did a video for better understanding of my problem. Thanks.

Video