Aren't these statements contradictory?
A change in an equilibrium system can shift the equilibrium position to the right,
Yes, by either changing the equilibrium constant (e.g., change in temperature) or by changing the reaction quotient under the same equilibrium constant (e.g., change in concentration of one of the species involved in the reaction).
Let's say that you have a reaction A + B
C, with an equilibrium constant of 2. You start out with equal concentrations of A and B and no C. Using the equilibrium constant you can determine the amount of C (and A and B) at equilibrium, which defines the extent of the reaction. If you then disturb the system by changing the temperature, you will have a new equilibrium constant, and the reaction will proceed to a new extent based on the new equilibrium constant. Likewise, if you remove some C from the equilibrium mixture, more A and B will combine to form more C, so that the reaction quotient approaches the equilibrium constant again. By continually removing C, you can make the reaction keep producing more C until all of A and B are consumed. This is a common strategy in chemical synthesis. But the extent of the reaction under a set of conditions is always determined by the equilibrium constant- if you want to change the extent of the reaction, you need to throw the system out of equilibrium in some way.