How can you say the average is unweighted if an average is price weighed?

An unweighted average means that you, the individual who constructs the average, do not apply any particular weighting. You simply add up all the stocks and then divide by the number of stocks.

But if you play around with averages for a minute or two, you'll realize that there is a naturally occurring weight among prices, one that you cannot avoid. Astock with a very high price will have a greater influence over the average than a lower priced stock. So, while a simple average has no special weighting applied, the very nature of numbers will bring its own weighting to the calculation. However, this natural weighting does not mean that a weighting has been applied to the average.