in drumming... all 4 limbs working independently from each other while still in concert with each other. Schooled drummers all learn that. I'm not a 'schooled' drummer... wish to hell I had been.
I will occasionally, (rarely), play hi-hat with my left hand and snare with my right just to develop independence. But not if it's a beat that requires concentration... because I'm not good at it.
On my acoustic kit I have two hi-hats.... an open and close hi-hat on my left, (traditional setup), and another closed hi-hat on my right... which allows me to be quicker around the kit. BUT... I seldom use both with my left hand as the lead the way you do.