Schizophrenic in the common (though admittedly mistaken) colloquial use of the term indicating near Gollum like changes in personality or attitude.
I'll defer to FilmCritHulk on this one, who laid it out pretty well even if I'm not much for his review overall.
I see, you mean the colloquial use. "Two personalities", or something to that end. Still not sure how that applies to Jon. That man lives on the bible belt. He wants to protect his son and he also wants to serve what he sees as the greater good. A man of conviction that even sacrifices his own life to that end. Had he not wanted to sacrifice himself in that moment, I'd be right there with the malcontents screaming schizo and inconsistent characterization as well. Replace Clark with a young Christ and Jon's actions might appear a little more understandable.
I personally think people are clouding the issue though a layer of preconceptions based on the many takes of the material that have come before.
Looking at Staker Pentacost and his various commands in this film, I don't see him too far out of step with the seemingly conflicting life decisions of Jon Kent. At the start of the film, this moral man orders the heroes to ignore the boat for the greater good(similar to the bus in MOS). Later he has his high scoring adopted daughter sit out fighting in the armageddon due to his sense of fatherly protection. Is this man interested in saving the world or isn't he? Perhaps we should have Hulk explain this to us.
As for what this Hulk character is talking about, I can't help but find things to disagree with him on, such as the dramatization of the father son bond. It can be found in the barn spaceship reveal scene "can I keep being you son" or the father son trope itself.
What's did clark learn form daddy's death? Glad he asked:
First off, Jon didn't tell his son not to help people, he said you have to keep that part of yourself a secret. If you notice, what he did on the bus needed a bit more of a discrete approach, where as what he was doing later was more low key and in drifter mode. Moreover, Jon also said keep your alien god presence a secret till the big joseph campbell calling scene presents itself and the world is ready, Jon knew his boy was sent here for a reason you see... Which brings us to the purpose of the tornado, the scene starts off with Clark in an ideological conflict with Jon, clearly this kid has his own ideas about how to live his life and is starting to doubt in his so called father's discretion, only to have his trust in his father fully cemented. A man that is willing to die for you is worth a good fathers day gift. A brash adolescent Clark walks away from that scene with perhaps a new found respect for his fathers beliefs, ones he puts into practice. The film could have added a few aftermath scenes to communicate this stuff further but I for one think there is something to be said for giving your audience some credit. Maybe too much credit in this case...but I digress, this is clearly the wrong thread.
The Prometheus example: The scientists were scared of the unknown in the cave. Then they happened upon a small snake that appeared non threatening. Sure they acted a little too forth coming, but these people weren't played as hollywood women who are afraid of their own shadow, if anything the could have been afraid of more humanoid zombie things with helmets, guns or xenomorphs even.
Replace the snake thing they found with a Newt and then suggest the premise was inconsistent. To trained biologists, alot of small creatures might not be as threatening as they are to you or I. Again, no reason for them to throw caution to the wind but to suggest people that were lost in a cave all of a sudden lost their consistency cause they found an animal that appeared friendly...obtuse I think.