Is the Isaac Newton Apple/Gravity Story True?

We've all heard some variant of this story: Isaac Newton is hanging around one day under an apple tree, an apple falls off the tree and onto his head, and he proceeds to formulate the theory of gravity. (After all, why would the apple fall down and not in some other direction?) This story has always sounded a little too convenient to be true; even Wikipedia has a lengthy discussion of the "Apple Incident" as part of its Newton page.
So did a falling apple really inspire Newton's thinking on gravity? Let's go to The Royal Society to look at some historical objects that tell us more:
This is the first video from Objectivity, a YouTube channel exploring the stories behind objects. I love it already. Here's a brief explanation of the channel from Brady Haran: