No need to be confused or see any ambiguity. The coin translates and rotates as it moves around. Translation is measured by the movement in the (x,y) co-ordinates if its centre. Rotation is measured by the angle theta a line that starts out going from the centre to the right-most point of the coin (imagine it drawn on) makes me with the radius that currently has the orientation.

X,y and theta completely define the position of the circle and are sufficient framework for thinking about the problem. Movements in theta form the revolutions of interest.