The general definition of a "coordinate system" says that the coordinates of a point specify a unique point. It doesn't say that a given point has only one set of coordinates. However, it simplifies matters if a point has unique coordinates. When a precise definition of "polar coordinates" is given, which is the case - do we allow a point to have many different polar coordinates or do we define them in such a way that the polar coordinates of a point are unique? Or is there some technical phrase like "standard polar coordinates" that indicates that such a definition is made?
A "polar equation" like r = cos(theta) is satisifed by pairs of numbers (r, theta) where r is negative. Do we say that the pairs of numbers that satisfy a polar equation "are" polar coordinates? Or do we only say that they are pairs of numbers can be converted to polar coordinates? (by finding a representation where r is positive).