/\/\I have always thought short to, long through, because the hitter is trying to hit the inside of the ball.
The first example from Phil Schonberg, is short to-short through, that's what I call T-rex arms, and it's not powerful. But I see a lot of players swinging like that in the warm up circle, and I cringe.
The second example from Phil Schonberg, is the short to, long through, extension after contact.
The second picture you provided isn't good to me either, maybe I'm reading into it, but it looks like that's what I consider casting.
The original question: Is extension thru the ball important?
Yes! Once you make contact, you have to give it a ride! (That's what I've told my DD since she was very little.) The hitter shouldn't stop her swing half way though.
And is proper finish a by product of proper extension?
Probably, but I'm just a softball parent, and most parents don't get that technical. Although, most of us know when a swing doesn't look quite right.