From the Post-Module Assessment 1A:

[Miguel Velasco 2J]

what is the difference between when a reaction is said to "Lie to the right " vs when the “equilibrium is shifted to the right” vs with respect to equilibrium constants?

[Jean Mok 3K]

To be honest I am not too sure about this exactly but from my understanding there is not much of a difference between the first two things that you quoted. I am not exactly sure about the third.

[Aijun Zhang 1D]

"Lie to the right" is used when describing K at the equilibrium. It is a state.
From the note, if K is small (K<10^-3), more reactants are at equilibrium. So the equilibrium sits to the left. Vice versa.
This basically describes which side is favored.

While "equilibrium shifts to the right" describes a dynamic situation. For a reaction, if more reactants are added, the concentration of reactants increases. So the equilibrium will shift to products (assuming products are on right side) to compensate the excess reactants, to minimize the effect of changes. This is more about change in concentration, pressure and temperature causing changes in the equilibrium shifts. The Kc is not changed during this process under the same temperature.

[Miguel Velasco 2J]

Makes sense thank you!