I feel that much of the tabletop and video game examples have lost the feel for what I see as a wizard was shown to be in the older literature. They were varying parts scientist, engineer, philosopher, and artist. Some worked towards mastering magic to understand the how and the why of the world around them worked, while others did it to impose order on what they viewed as a disordered world but for whatever reason it is often a compulsion at least on an intellectual level that was impossible to ignore and that they understand and control this force and it's under laying rules in the world around them.
Where this differs from a Sorcerer is that while a wizard is studying and learning about magic a Sorcerer has it in their blood and bones either through birth and heritage or from strange accident, they can no more get away from it than they can get away from themselves. The side effect of this is it tints and bias their interaction with it and limits breadth or range in how they can interact but this gives them the advantage in that while they lack breadth they more than make up for that in their depth of understanding because of their natural intuitive understanding.
On the interpretation of the Wizard schools, what I would like to see, but I'm not really expecting it to have schools more refined the character than entirely define the character.
Such that possibly a School on Battle Magic might give some training in a limited number of martial weapons or something in that vein. Note Battle Magic, in my opinion, should not be defined by one school anyway, being as a conflict has many sides that could be supported with magic (seige craft, troop movement, espionage, control of terrain, etc)
This is just my take on