If you aren't showing, then the best grip will be the one that is most secure but still allows for softness and fluidity in your hands and arms.
Personally, I cannot fathom how people can hold their reins like this, even though I know a lot of people do.
When I'm riding 2 handed, I always have my reins crossed over the neck and have both reins bridged and held in my hands like this. Also, you'll notice in this picture that I have one pinkie below my reins and one splitting my reins. It may not be "proper", but that's how I have the best feel for the horse's mouth.
And please forgive the puppy paws, it was a picture taken at a wonky moment LOL.
As for the straight line thing, it's generally considered good horsemanship in my area to keep your forearms pointed directly at the horse's mouth even if you don't have contact on the reins. *looking down is one of my many bad riding habits, especially on greenies
Now, because I start training neck reining within just a few rides of them being backed, I usually start riding in only one hand and use the other for corrections only. When I start this, I keep the reins bridged across their neck and I hold them like this because I can angle my hand to give direction with the leading (direct) rein. I can pull with the pinky side of my hand to go left or I can pull with the thumb side to go right)
Then, when they are neck reining well and I no longer feel the need to bridge my reins, I hold them like this with both tails on the left side of my horse