If you wanted to be totally unambiguous, “the k-nearest neighbor classification algorithm” might be better, but it’s pretty common to abridge that to “nearest n

If you wanted to be totally unambiguous, “the k-nearest neighbor classification algorithm” might be better, but it’s pretty common to abridge that to “nearest neighbors” or “KNN” in a machine learning context. You are absolutely right that there are many ways to efficiently find those neighbors, but the classification algorithm itself doesn’t care; it’s implementation detail.