Yes, it's definitely close enough to quicksort to classify as that, or at least a minor variant.
You are partitioning the data, you just don't have a separate function for it.
You do have something a bit strange happening.
Normally, with the in-place variant of quicksort, you increase the left index while it's smaller than the pivot, then you decrease the right index while it's larger, then you swap the two (as to only perform swaps involving 2 elements which are both on the wrong side).
You are, however, just increasing the left, so you could swap already larger elements from the right. This could result in a few more swaps than necessary, although the asymptotic running time should be the same.
Here's some pseudo-code of the 'normal' variant: (taken from Rosetta Code)
while left ≤ right
while array[left] < pivot
left := left + 1
while array[right] > pivot
right := right - 1
if left ≤ right
swap array[left] with array[right]
left := left + 1
right := right - 1