Difficult cold start

I am confused what causes this problem as all seems to work (ignition, carb, starter), so I am suspected for all components.

Don't fall into the trap of just throwing random lists of parts at the problem. That way you spend a lot of money, time, and effort, and likely don't fix the problem, likely introduce new ones, and never figure out what was causing the problem.

I have no reason to suspect starter. Just have experience from other vehicle where new starter with "transmission" (I don't know what word in english means that starter has gears and it is not direct driving flywheel) made the difference.

Your original starter is a geared one. So is the mini starter.

Yes, there is something wrong with engine if it does not start with working original starter, but it would work as workaround maybe?
No.