When 3+ hexaaqua ions react with hydroxide ions, the following set of reactions occurs:
When we get to stage 3, a neutral ion has formed, and a white precipitate is created. Now, we know that for amphoteric species, adding $\ce{H+}$ ions will reverse this process, hence removing the precipitate. Surely this is an effective test for amphoteric nature?
However, according to my exam board, if you add an excess of hydroxide ions (hence forming a negative ion) after initially adding a small volume of hydroxide ions, and then the precipitate dissolves, this proves that the metal is amphoteric. This does not make sense to me because the definition of an amphoteric species is one that reacts with both acids and bases; yet, in this test, the species has only reacted with hydroxide ions; no acid - so there is no proof of amphoteric nature.
So, why is a species amphoteric if it forms a white precipitate after reacting with hydroxide ions, then dissolves after adding an excess?
An example of a problem is: "how do you prove lead (II) hydroxide is amphoteric, whereas magnesium hydroxide is not?" The answer is: add an excess of $\ce{NaOH}$ - magnesium forms a white precipitate, whereas lead(II) hydroxide forms a white precipitate, then dissolves. There is no mention of acid, so how does this prove amphoteric nature?
If anybody can clear my doubts or explain where I am wrong in thinking, I would be very grateful.