For some reason, it’s common for Americans to mention that the USA isn’t a democracy but a Republic. I’m not aware of any other country where such a statement is made.
“Democracy” and “Republic” aren’t antinomic. They refer to two different things. The USA is a democracy and a Republic, Syria is a Republic but not a democracy, Spain is a democracy but not a Republic, and Saudi Arabia is neither a Republic nor a democracy.
If you just say to someone who never heard of the USA that it’s a Republic, he can’t guess what your political system is. If you then add that it’s not a democracy, then he’ll rightly assume that it’s a dictatorship of some sort (or possibly an oligarchy).
My understanding is that when it is said, what is meant is that the USA isn’t a direct democracy. But then, there aren’t any direct democracy on the planet anyway.
ETA : the definition given in your link for “Republic” is wrong from the get go (“Republic. That form of government in which the powers of sovereignty are vested in the people”)