The trick here is actually the solubility of the solute.
As you know, a saturated solution is a solution in which an equilibrium exists between the undissolved solid and the dissolved ions.
In other words, a saturated solution will contain the maximum amount of dissolved solid at a given temperature. This maximum amount is the solubility of the salt at that temperature.
Now, a diluted solution is simply a solution that contains relatively small amounts of solute compared to the amount of solvent.
Consequently, the term "diluted" does not carry any meaningful information about just how much solute is present in the solution, it just tells you that you have "much" more solvent than solute.
So, let's say that you have
In this case, the solution will be saturated and concentrated because it contains the maximum amount of solute that can be dissolved at that temperature, i.e.
On the other hand, if you take a salt that has a low solubility in water at that temperature, let's say
In this case, the solution will be saturated and dilute because it contains the maximum amount of solute that can be dissolved at that temperature, i.e.
So remember, saturation is an absolute term because it depends exclusively on the solubility of the salt at a given temperature.
On the other hand, concentration/dilution are relative terms because they depend on the relative amount of solute present compared to the amount of solvent.