Err, Fate point isn’t “Did you touch every facet of this belief this session, except for completing it outright?” It’s “Did this belief noticeably drive play?” It’s okay if it’s only “part” of your belief, so to speak. (It is worth avoiding colossal grab-bag beliefs for other reasons, though. The structure is there to encourage you to pare stuff down to a manageable set of priorities. Sometimes it will be super-hard to actually do that. I think your example of “teach, protect, &c.” is fine, though. That’s just defining your role in this kid’s life more thoroughly.)
[Edit: Scooped by Alex.]
Fate: “This award is given when playing a Belief serves a purpose and drives the game forward.”
It doesn’t have to be every single element of a belief. You don’t even have to accomplish any part of the belief. You just have to play towards the Belief and have it drive the game. Do you put effort into teaching? Fate. Go to lengths to defend the prince? Fate. Pursing pieces of a belief is valuable and gets rewarded. Obviously it’s cheating to cobble together a belief out of entirely disparate parts, but you should never be punished for being detailed and explicit.
What if the Belief were condensed to “I will nurture the prince into a ruler.” That’s simple and succinct. Protecting him probably counts as nurturing; you can’t nurture if he’s dead. Teaching is nurturing. So is making sure he’s fed. But that’s a blander Belief. I don’t like it as much. I would never, ever give less artha to someone for breaking down a belief into clearer parts with actions to pursue.
On the other hand, if your belief is to teach and defend, you’re on an unrelated journey, and you teach the prince on the road and defend him from bandits who attack… no fate point. Why? Your belief was incidental. Sure, you played it, but it didn’t do anything. You were journey, you did stuff that had to happen. Your Belief didn’t move anything forward.
Ha ha too much good advice 
Think I’ll leave it where it is now though and develop as I play. You’ve all been great help