If I were learning to play from scratch with what I know now about playing the drums I'd learn how to play open handed. That would allow me to play with my right hand on the snare but I'd also learn to play with my left on the snare as well. Doing both strengthens the amount of independence you can exhibit at the kit, which just makes all sorts of things so much easier. Unfortunately I'm an old fart and it would take me too much pain and suffering to learn open handed playing. Watch Carter Beauford of the Dave Matthews Band. He's insane and plays open handed. Many others as well.
Your preferred hand or the one you write with. Most natural. I do suggest learning to lead with both left and right hand, it will pay off later that your limbs are much more independent from each other
With the drum set being a little over a hundred years old it's a new born compared to other instruments. That said tradition is a blink of an eye. Be versed with both hands. Look up Simon Phillips on you tube.
It all depends on if you have a right or left handed snare drum.
When I had to start all over, I really wish I had focused more on playing open handed (right hand on snare), as a mentor had taught me, but the whole left side of my body was slower to recover and I wanted to get back into it ASAP.
Open handed is a great way to develop your left hand which is typically weaker for a right handed individual. As others have said, learn both.
Edit: I didn't see MntnMan62's post above. But, yeah, what he said.