Thanks again to everyone!
I've narrowed my problem - it starts if i play with gears for a bit before trying to start a bike (First->Neutral ->First->Neutral->Second->Neutral ...) Right now it is obvious that's a problem with some connection, but I do not know where it can be. I have checked battery, switch and some other clamps, they look ok, not rusted or anything like that. They are still cleaned and connected back - nothing has changed since
