There are a few questions to ask when dealing with large quantities of data (I leave "large" intentionally vague).
Can all of the data fit into memory at once?
Is the amount of data known up front? Or could more come in while the algorithm is processing?
Is it feasible to sort the data before analyzing it?
Most college-level exercises tend to deal with small amounts of data, so these questions do not matter. However, consider Google: they have massive amounts of data, and loading the entire search index into memory and sorting it is nowhere near feasible.
Even when the amount of data could fit into memory, sometimes the best solutions are ones that work in the larger case. For example, you may not need to sort this array to find the largest elements. From a theoretical perspective Big-O notation is great: however, in the real world, sometimes the fact that O(n + n log n) simplifies to O(n log n) is not a big reassurance. It could translate into minutes or hours of extra run time.
While the link you posted claims it to be O(n), as does Wikipedia, I disagree. Big-O is about worst-case complexity, which is actually O(n2). While the average-case is n (there is no letter for that), the worst-case is a huge time killer.
The array is iterated once, which is O(n). Modifying the binary heap is O(log2 k), but is performed (worst case) n times. This makes the whole thing O(n + n log2k) = O(n log2k). Also keep in mind that k is small compared to n, and log2k is even smaller.
Partial selection sort in this instance is O(kn).
The first option may in theory perform well most of the time, with certain data sets performing poorly.
The second option will perform very well all of the time. In the worst case, it is much better than the first algorithm. The first algorithm may outperform this one some of the time, however.
The third option is pretty much constant for a given n: the contents of the array do not matter, only its size.
I would lean toward the binary heap myself, but it might be worth analyzing whether the data would give the Quickselect algorithm a difficult time and perhaps implement both and measure their real-world speed.