I'm trying clear up some misunderstandings about the term "republic", and especially its relationship to the concept of democracy. Note that my definition of the term contradicts Webster 1913, but I believe it to be the more correct one.
In order to properly define "republic" I'll analyse where the world comes from: "Res Publica" - Latin for "the public matter". The idea of a republic then is that all matters are divided into the public and the private sphere.
What matters fall into which sphere depends on the actual state we're talking about. One can generally agree that the colour of your shirt is a private matter, whereas how fast you are driving your car is a public one. This question becomes more difficult in other areas: Can one legislate morality? (sex laws, prohibition) Can one legislate political expression?
Independent of where the dividing line is drawn, the identifying feature of a republic is this distinction in between public and private. This also has consequences for those in power: The president of a republic is simply a person currently in the office of the president. In his* job as president he is part of the public sphere, but when he goes home in the evening, he becomes just another private human being like anybody else.
This contrasts with the idea of a King. A King does not serve in the "office of the King of x". The King is the king. There is no distinction in between the public and the private person of the king.**
Moreover, a republic does not have to be a democracy. Most republics are, but an aristocratic form of rule is an alternative, or really any ruler at all. The concept of a republic does not concern itself with who rules, only with what they rule over - the public sphere only.
* Or "her job", of course. I just dislike mangling language.
** Feudalism is a matter of personal relationships. Everything is private, the concept of "public" does not exist.