Of course they don't see the strike zone the same way. That said, it is very difficult for a hitter to see a pitch all the way to the strike zone. At some point the low pitches can literally disappear on hitters. In fact, the last actual view a hitter has of a pitch is some number of feet out in front of the plate. The human brain is simply not capable of processing information from the eye quick enough to "see" the ball all the way to the strike zone. Besides, the hitter cannot possibly start his swing at the instant the pitch is crossing the plate. His brain is processing whatever information his eyes sent to his brain when the pitch was several feet out in front of the plate and using that to guide his swing.
The closer the ball gets to the plate the more it is simply a blur to the hitter from his perspective astride the plate. The hitter is "calculating" where the blur will be when his bat is in the hitting zoned translating that to his body thus controlling his swing. That is the best he can do. This is also why the uppercut swing is so ineffective compared to a level swing, which is what X-man and Rafi have. The uppercut swing simply does not stay in the hitting zone long enough and it compounds the problem of the ball being a blur to the hitter who is simply guessing as much as anything else. The uppercut is why there are so many K's now. If you hit it the ball goes a long way in the air. But the chances are greater that you won't hit it at all or won't barrel it up.
The umpire and the catcher actually have the best view of pitches and of those two the umpire has the best view of all. He is only trying to call the pitch. The catcher is trying to catch it or block it and that consumes a good deal of his concentration. However, Umpires do not call strike zones consistently. Whatever box is put up on the screen is provided by way of camera and computer. Whether it is "accurate" or not, it is consistent. It is in fact the only "consistent" functioning ball and strike observer of all of the participants in the drama we call baseball. None of the observers of balls and strikes are entirely accurate and some are not even close to accurate. Most of them are nowhere near consistent, not even close to consistent.
The worst view of the pitch..... everybody else watching. The worst of the worst, those yammering from the dugouts. The best of the worst is the hitter's view. He is using post processing information stored in his brain to make HIS call of the pitch.