What is the optimum design: elliptical lift distribution or elliptical span loading parameter distribution? I am getting very confused with these terms. I'm trying to analyse a span loading  distribution given as cl(y)c(y)/ccL. This distribution is almost elliptical but when I plot just cl(y) it changes and I have more cl near the tips due to the reduction in chord.

This made sense from my point how view since the AR is low, the sweep is high and the taper ratio is small. It is like the figures I remember for these cases. However, when I have searched in google I have seen this image:


This wing is pretty similar to mine, and the distribution is similar to my cl(y) distribution. However, the parameter here is cl(y)c(y)/ccL!! 

I don't understand completely the difference. When we talk about elliptical distribution for minimum induced drag it is about cl(y) or cl(y)c(y)/ccL? Why is this parameter used and plot and not cl(y)?
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.