Defining the position of equilibrium I've read quite a few other answers on this site such as this one, but can't quite seem to understand fully yet.

Say we have a container with gases reacting such that the number of moles on both sides of the equation are not equal. If we then, for instance, reduce the volume of the container, the total pressure increases and the value of the quotient Q will also change. The net reaction then favours a particular direction so that the mole fractions change sufficiently for Q to return to the value of K, which remains constant.

This is commonly described by saying equilibrium shifts to either the left or the right. Whilst it is true that after the system has again reached equilibrium the mole fractions have changed (so in a sense, the new equilibrium has shifted to one side in terms of number of moles), the value of Q is equal to its initial value so in this sense the position of equilibrium as defined by K has not changed.

The same applies to changing the concentration of a species; the actual concentrations of every species after the system has re-equilibrated will be different, however the value of Q will have returned to its previous value. 

The only situation that I can think of where the position of equilibrium actually shifts is when a change in temperature alters the value of K.

My question is, essentially, is the term 'position of equilibrium' more of a qualitative descriptor of the composition of reactants and products? I initially thought it meant the value of Q at equilibrium (or K), but evidently in gaseous reactions the equilibrium yield does change with changes in pressure though K remains constant, so such a definition would not work.

Thank you!
"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).