When I select "Duplicate particle systems" a copy of the first particle system is created. This copy can be edited in Particle edit and I can even add new particles there. But if I change any settings (for example, change Children, Root, Material & etc.), then these settings also change in first particle system. How can I duplicate a particle system so that I can edit the copy, but that this doesn't affect the first particle system?
you have to play with system block and data block so you don't have the same particle system applied twice. Probably you'll need to click on that "files" icon (that let you duplicate the particle system, starting from a cloned version of the original one)
This is correct:
check how to work with different emitter settings:
this is wrong (see the number 2 that means that the particle system is used twice)