What Measure of Disorder to use when Analysing Quicksort
Lomuto vs Hoare
Lomuto partition suffers when sorting equal keys, whilst Hoare partition does not.
Both partition schemes suffer equally when using a pivot distant from the median.  

Measure of disorder
The measure of disorder to choose for the purposes of quicksort is simple.
A: How far removed from median is the fixed pivot, compared to random data?
If you insist on using Lomuto partition and if you assume duplicate values are allowed you need to add the following test against randomness:
B: How many duplicate elements are there, compared to random.      

Of course it is rather silly to assume that duplicate values are allowed in your data set and still evaluate Lomuto partition, so you should probably either eliminate duplicates beforehand or switch to Hoare partition or assume duplicates are rare.  

Both measures are trivial to quantify using statistics.  

We can rule out pathological data
Any other deviations from randomness will not matter for the purposes of analysing quicksort. As long as the pivot is close to the median it will perform well on all data that is not pathological.
The distance from random would have to be great indeed to be quicksort-pathological, so we can rule that out.     

Never use any fixed pivot(s) in real code
Do note that if you write real code with a fixed pivot*) (whatever that pivot may be) you are opening yourself up to a denial of service attack, because an attacker can insert a pathological value at just that point and thus you should always choose a random element as pivot.

*) or multiple pivots if you choose best of x pivots.