Hey, the new Ford ain't nuttin like the original. It's reliable, quiet, fuel efficient, and has an electric starter. We still call it a Ford with pride, though.
The question is whether Tesla defined his coil in such a way that would exclude these later derivatives? Clearly, the little tuning knob on the end makes quite a difference. Did Tesla ever patent his coil (I am not a student of Tesla)? If he did, that would be a good source to see how broadly he considered the concept.
John
Edit: Here's one of Tesla's patents:
http://www.tfcbooks.com/patents/1119732.htm The fact that adjustment of the oscillator changes the output (the effect is quite noticeable when looking for pinholes) would seem to qualify it as a Tesla coil. I'll bet Tesla's attorneys would also agree -- at least while the patent was valid.