Defining the position of equilibrium
"Shifting" an equilibrium

Unfortunately for learners of chemistry, statements like "the equilibrium shifts to the right" are quite common jargon.

What is meant by this, speaking in more accurate technical terms, is the following: A reaction had reached equilibrium (net reaction is zero), and then some change occurred so that it was no longer in equilibrium. Now, there is a net forward reaction in the direction of the products ("to the right" refers to reactants turning into products).

"Shifting to the left" describes a similar sequence of events, just in the reverse direction.

New equilibrium or reestablish equilibrium?

The OP is correct in saying that unless the temperature changed, the equilibrium constant does not change, and the reaction quotient Q of the first equilibrium state is equal to that of the second. So you could say you are back to the same equilibrium.

Typically, however, the mole fractions, concentrations or partial pressures will be distinct for the first and the second equilibrium. So you could also say the reaction reaches a new equilibrium.

If you say it reaches equilibrium again, you are sufficiently vague to be correct.

Position of equilibrium

The term "position of equilibrium" is ill-defined. To avoid any confusion, it is better to distinguish between reaction quotient (which is equal to the equilibrium constant for both the first and second equilibrium) and the set of concentrations (which is typically different).