"The design lift coefficient at which the airfoil section produces the least drag." and "The lift coefficient of the airfoil chosen so that it matches the airplane lift coefficient." Are both consequences of the definition of the design lift coefficient -- not its definition.
Imagine a thin flat airfoil (zero thickness) at positive angle of attack. We know the rear stagnation point occurs at the TE because of the Kutta condition.
The forward stagnation point occurs aft of the LE on the lower surface. The streamline immediately above the stagnation streamline will turn to flow up the airfoil, will turn 180 degrees at the LE, and back down the upper surface to the TE. At the LE, when the flow turns 180 degrees at a point, the acceleration is infinite and a large pressure spike.
When at negative alpha, the same thing happens, but upside down. The forward stagnation point is on the upper surface, flow races around the LE causing a big pressure spike.
Between these phenomena, there is one alpha where the oncoming streamline perfectly meets the LE, and the forward stagnation point is exactly at the LE. In this case, there is no need for infinite acceleration and the resultant pressure spike.
For a flat plate airfoil, this happens at zero degrees alpha and at zero lift coefficient.
For a cambered airfoil, this happens at some alpha (maybe zero, maybe not) and at some (most likely non-zero) lift coefficient.
The lift coefficient where the forward stagnation point exactly meets the leading edge of the mean camber line is the ideal lift coefficient for that camber line, cli.
That is the definition. The rest are consequences.
When you go to thick airfoils, the infinite acceleration goes away. There will still be a substantial acceleration with a pressure peak -- but it is all finite.
Operating a thick airfoil at cli will reduce accelerations around the LE and will reduce pressure peaks.
High local velocities tend to increase the profile drag on an airfoil. Strong pressure peaks require stronger adverse pressure gradients to traverse without separation.
Assume you want to operate an airfoil at cl=0.3. One is a symmetrical foil (cli=0.0) and the other is cambered such that cli=0.3. They have the same thickness distribution and the same LE radius. Angle of attack of both airfoils is adjusted to meet cl=0.3.
The cambered airfoil has an 'easier' time achieving the desired lift. It can do so without excess supervelocities around the LE and without a strong pressure peak. It will most likely have the lower profile drag of the two.
Because of all this, the cli is usually located near the bottom of the drag polar for a conventional airfoil -- and near the middle of the drag bucket for a laminar flow airfoil.
If you want to operate a wing with chord distribution c(y) at a target CL with a desired load distribution (cl(y)*c(y)) (perhaps elliptical, perhaps not), you can determine the target cl(y).
This sectional lift coefficient distribution cl(y) achieves your induced drag goals balanced against your structural goals.
By choosing a spanwise distribution of airfoils that match cli to your target cl(y), you choose airfoils that will each have an 'easy' time doing their job. They will each operate near their lowest profile drag condition. This will tend to reduce the reduce the parasite drag of the wing near this design operating point.