Strictly speaking do the Hoare and Lomuto partitioning algorithms work on the same algorithm: quicksort?
The term "Quicksort" stands for this abstract algorithmic idea:


Pick a value $x$.
Partition the input into $\{y\mid y \leq x\}$ and $\{y \mid y &gt; x\}$.
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 for a discussion about the differences of these particular two variants.