The term "Quicksort" stands for this abstract algorithmic idea:

 1. Pick a value $x$.
 2. Partition the input into $\{y\mid y \leq x\}$ and $\{y \mid y > x\}$.
 3. Recurse on the partitions (if they are non-trivial) and append the results.

You may want to generalise to multiple pivots, or create a third partition for elements equal the pivot, but mostly that's it.

There are many, many implementations. All of them are called "Quicksort", but many modify the name.
See [here](https://cs.stackexchange.com/questions/11458/quicksort-partitioning-hoare-vs-lomuto?rq=1) for a discussion about the differences of these particular two variants.