Battery replacement problem, mac does not turn on after discharging

Let’s get a better idea what’s going on. Download this gem of an App CoconutBattery with the MagSafe charger plugged in take a Snapshot of the Apps main window with the Battery tab opened up and post it here for us to see.

@lukaszartemiuk - OK, while the batteries cycle count is very low, the age of this battery concerns me! It’s over five years so it’s chemistry it not at its prime. So it likely needs a new battery soon.

At this point your system should work, so something else is effecting it. From your description it appears SMC is going into a latched condition not a battery issue.

Let’s do one more test here, restart your system and press the D key to enter the onboard diagnostics. See if it reports any errors let us know what you get.