What is the optimum design: elliptical lift distribution or elliptical span loading parameter distribution?
Minimum induced drag is reached when the circulation distribution over span is elliptical. Circulation is indeed proportional to the product of the local lift coefficient and the local chord (c$_l \cdot$c). The diagram in your question normalizes this with the overall lift coefficient c$_L$ and the mean chord $\overline{\text{c}}$.

But this is only true if you disregard the structural weight of the wing. Using less load near the tips will reduce the wing root bending moment and, consequently, wing mass, so less lift and wing area are needed in order to carry the same non-lifting load. Even though your drag coeffcient might be higher, your overall drag will be lower.

The optimum will also depend on the size of the wing. Small wings should indeed use an elliptical circulation distribution because their structural weight is low, but in large airplanes a more triangular distribution will be better.

One last word: Please try to avoid the high loads near the wingtips if you want a nice, docile stall behavior.