MOSFET does not turn off totally
I bought some N channel MOSFETs (IRLZ44N) - which has logic level gate drive from a reputable local store (for about 0,36EUR / 0,43 USD per piece).
According to datasheet, for 3V gate voltage and 3A drain-source current the resistance goes below 0,1 Ohm...

I changed the N channel MOSFET (and only that) - and VOILA everything works like charm! As I already connected to PWM signal, at even 1% PWM LEDs are nicely (but barely) dim.

Now I am sure that the originally bought IRF MOSFETs from eBay are fake (IRFZ44N were 1 USD/5 pcs; IRF9540N 1,22 USD/5 pcs) - so a price tag difference was (the real in Hungary, including 27% TAX) about 0,23 USD per piece - I start to pull my hair out.... - and always check local sellers next time.

I also bought P channel MOSFETs (IRF9Z34N) and I am quite sure my other circuit would also work with a real MOSFET.