L13 question about battery

I have a L13v4 but since I typically use it as desktop replacement I have unplugged the battery.
The last month has the following behavior: When I plug it in the mains and press the power on button the two leds light together and the screen remains black (does not turn on). Now I keep the Power pressed and the laptop turns off. I press again the power and now one led lights, the screen turns on and the second led light after that. Everything works fine.

This repeats every time. If I leave the laptop always plugged in the AC outlet, and turn it on, it works immediately. But if I remove it from the mains, reconnect and try to turn on the procedure repeats.

So I guess that if the battery was internally plugged I would never see this behavior.

My question is, maybe some motherboard battery needs to be replaced? (the clock is correct, no problem there).

This is somewhat tangential, but my L13v3 recently refused to boot and I ended up taking off the back and unplugging/replugging the battery connector and the CMOS battery. My L13 then commenced to boot properly and all is well.
Perhaps when you leave your laptop off mains, the CMOS battery becomes discharged and your laptop resets.
After you get your laptop rebooted, check the date and time. If incorrect, your CMOS was also reset. You might want to plug in your main battery and replace the CMOS battery to get back to normal.
To avoid battery wear, I keep my laptop plugged in but every few days I unplug and discharge it down to 20%, then recharge it.

most likely the EC simply doesn’t like being without power, and takes an additional reboot after power is reapplied for it to actually boot

This is OK, but it is happening now after 2 years I have being using L13 the way I do. So I guess I have to change the motherboard battery. Is there only one such battery inside?