How is a vessel that contained pure water and became defiled by drops of impure water purified? Is it sufficient to pour out the defiled water, then pour a small amount of water and swirl it around, or must it be placed under a tap to be filled with running water?

This question stems from waswas (whispering doubts), and the best treatment for it is to ignore it. If drops of impure water splash into a container with water, and the water does not change [in color, smell, or taste], then it remains pure according to some scholars, and the container remains pure. There is no harm for someone afflicted by waswas to adopt this view. However, if you wish to follow the view of the majority, which holds that a small amount of water becomes impure upon contamination, then for the purification of the container, it suffices to have water flow over it, as long as the impurity has no physical substance [like color, smell, or taste].