The information comes from my experience using the GE, and what others have told me.
Players control prices through the GE because all offers in the GE are placed by players. The GE "median" price might be off sometimes due to the minimum trade requirement for it to be updated, and the fact that street prices sometimes swing wildly. Case in point: partyhats. White partyhat is like 1880m now, and the GE price is waaaay under that.
If items were being purchased from Jagex this would severely oversupply some markets such as third age or discontinued items. It's common to see people buy and sell items from each other through the GE in R Quark/Nex and SS; I have no evidence atm that any items in the GE are purchased from Jagex.