Well... I know there's actually three ways to hold a traditional mouse, I saw a video about it once. Which was surprising to me, but mostly because I have big hands and I can only hold a mouse one way.
But for me when I'm holding a horizontal mouse the mouse is "grasped" by my hand as a whole, and I'd say my pinky and thumb are in a way holding the mouse.
With a vertical mouse it's much the same where my whole hand is grasping. I'm not holding it any more with my thumb because the thumb rests on the body of the mouse. In fact as I think about it and type this out I think I have to hold a horizontal mouse more than a vertical, but again it might just be related to hand size.
I use both. Horizontal for gaming so I can have more than two thumb buttons. Vertical for work as it does give my wrist and forearm a "rest."
Edit: realized I didn't necessarily answer your question. For me with the vertical mouse the buttons are just there where they need to be. And I guess clicking is a form of holding...
Imagine your favorite thing in the world - sweaty men trying to push a handegg up and down a field - being infiltrated and rigged so even that is no longer sacred for your Sunday winter ritual. Fucking epic.