Are cerebellar basket and stellate neurons actually different cell types?
You are completely right with your guess that is was Ramón y Cajal who first described these cells in his famous monograph "Histologie de systeme nerveux de l'homme et des vertebres." published in Paris Maloine in 1911. 

Since the only method he had at his disposal was Golgi's staining which is a pure anatomical staining (there is no living cell left in the tissue after the treatment with  potassium dichromate and silver nitrate) with silver chromate filling completely single cells at random.

Using only this staining and his eyes Cajal did classify the neurons according to their size, location in the brain and their shape. This is why we have such names like basket, stellate, moss, pyramide etc. in neuron description. This classification is purely anatomical and does not reflect any intrinsic properties of these cell, as for example being motor neurons or interneurons.

When I was studying histology at my medical school I was always told that there is new "mollecular biology"-based classification of neurons is about to come and to replace the existing classification by Cajal, many cell types were supposed to be merged there, among others also different anatatomical groups of interneurons. (This was one of the major research topic of the research facility where I studied neurohistology).

As far as I can see, this classification is still not here...