I would think that the original Gygax characterization of roleplaying being a function or responsibility in a playing group is most accurate, but I would never be adamant about that. My own gaming group does very little "personality" role playing, the class and race of their character being the primary definition of their character. So much of the rule book is based on the mechanics of a class and the various ways to flavor the class with feats, spells, weapons, magic, etc.
However, the roleplaying of personalities is a huge part of the game, it is why players write histories and backstories for their character. It is critical to developing adventures that matter to the character. Without the personalities, the game would probably be a bit boring and all the adventures interchangeable (I'm sure other DMs would agree that not all adventures are suited for all adventuring parties). It is interesting to note that I think even WOTC recognized that personality roleplaying may be a weak spot for players when they wrote their recommendation for using the DM's Inspiration Die: "Inspiration is a rule the Dungeon Master can use to reward you for playing your character in a way that’s true to his or her personality traits, ideal, bond, and flaw. Your DM can choose to give you inspiration for a variety of reasons. Typically, DMs award it when you play out your personality traits, give in to the drawbacks presented by a flaw or bond, and otherwise portray your character in a compelling way." Sure other ways can earn inspiration, but I doubt many DMs are handing out inspiration to a wizard for casting a spell or to a fighter for swinging a sword. I know the "Inspiration Die" is much maligned, but its connection to the personality roleplay suggests that the personality aspect of roleplay is not required as much as is the function of class in the game, but the personality aspect of roleplay certainly does make the game much more fun.