Can somebody briefly explain why eye dominance is important?
When it comes to handguns it's not important.
I'm left eye dominant, but shoot handguns right handed. I shoot long guns lefty so that I can still focus with both eyes open.
You will get a lot of comments about this from people that are NOT cross dominant. Ignore them.
Please bear with me while I tell you my story.
The first time I shot a BB-gun, I instinctively wanted to shoot lefty. Unfortunately the narrow minded folks I was shooting with insisted I shoot righty because I did most other things right handed. I struggled to shoot well.
When I joined the military and started shooting handguns they insisted I shoot right handed and use the Weaver stance. Again, I struggled to shoot well.
When I sought private instruction for shooting handguns, they made a huge deal out of cross dominance, and had me doing all sorts of wacky things. Again, I struggled to shoot well.
What did all those instructors have in common? None were cross dominant shooters themselves, and all made the issue of being cross dominant out to be a huge a handicap.
Guess what it's not a handicap - AT ALL.
Simply shoot long guns left handed if you're left eye dominant. You can't change eye dominance, unless there is some drastic change in the vision of one of your eyes, regardless of what some might tell you. My source on that is two eye surgeons.
However, training someone to handle the typical shotgun or rifle with their "weak hand" is relatively easy. By shooting the long guns on the dominant eye side the dominant eye lines up with the sights, and you can keep both eyes open.
With handguns just shoot with whichever hand you like. Unlike long guns the sight line of the handgun can easily be lined up with either eye. Shooting Weaver with an extremely bladed body will make it difficult, but Isosceles (or modified Isosceles with a slight blade to the body) makes it easy to line up the gun in front of the dominant eye with either hand. Isosceles/Modified Isosceles is what the top competitive shooters are using anyway.
Please note that a lot of instructors, who are not cross dominant shooters, will make a huge deal about teaching shooters who are cross dominant. It's BS. Notice that when instructors talk to shooters, who are NOT cross dominant about shooting "weak" handed, they will talk about all kinds of things a shooter will need to adjust for, but don't EVER mention cross dominance.
If it's not a factor for a non-cross dominant shooter shooting "weak" handed, why do they insist on making it into a big deal with shooters who are cross dominant? The answer is it's not a big deal. Just bring the pistol up in front of the dominant eye, and go.
When I was finally taught to shoot long guns lefty, and simply put the pistols in front of the dominant eye no matter which hand I was shooting with, I quickly went from struggling to be a decent shooter to being a very good shooter.