In 'What is Democracy?', I argued that majority rule may, in certain circumstances, constitute a form of oligarchy rather than democracy. To truly qualify as rule by the people -- rather than some (however large) subset thereof -- it must be the case that anyone has the opportunity to significantly affect the outcome. One vote alone cannot do this, but if each voter is receptive to being persuaded by any other, then we each have the potential to persuade a large number of people (if we come up with a sufficiently powerful argument, say) and thus to exert significant political influence. I thus concluded that 'democracy' in the fullest sense is realizable in the form of deliberative democracy.
But it's worth noting that this is not to conflate 'democracy' and 'good governance' (as emerged in comments). Although deliberative democracy would ideally involve citizens being responsive to (good) reasons, it's possible to invert the normative substance whilst maintaining the democratic form. That is, we might have a citizenry that is receptive and reliably responsive to bad reasons. All you've got to do is come up with a sufficiently atrocious argument, and you can exert significant influence. This anti-rational society is surely democratic in the strongest sense, being formally identical to the ideal deliberative democracy. But it is a (substantively) terrible government nonetheless!
Another possibility is that a democratic society may be responsive to some other universal capacity besides reason: perhaps rhetoric, or humour, say. Again, these needn't be good political orders, but -- so long as each citizen had the opportunity to exert significant influence in this way -- such a state would presumably qualify as "ruled by the people" in the strongest sense.
These examples suggest that democracy is not strictly sufficient for political legitimacy. There's no reason to accept or abide by the rules of a reliably wrong, anti-rational system, and it's at least logically possible that a democracy might be unreliable (or reliably atrocious) in this way. I'd suggest that whether a government is legitimate is more a matter of whether it is sufficiently reliable and responsive to good reasons.
In short: democracy requires that citizens be receptive to each other's persuasion (somehow or other). Legitimate democracy further requires that what the citizens are receptive to being persuaded by is good reasons.