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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 for a discussion about the differences of these particular two variants.