Stirling engine

Where [itex]V_2>1[/itex]. It's also derived here.
But my ("Highschool", or secondary school) teacher says that it can't be expressed that way, and is therefore wrong. Because she is considering how the Stirling engine works in practice. She says that because the heat that flows OUT of the system, from the isochoric proces, can be "stored" in some sort of heat storage device (the last animation here, called regenerator), then you don't need to take the positive heat flow TO the system, into account, when calculating the efficiency. Because that energy is stored, and doesn't need to be heated from the fuel. By this way, the efficiency of the stirling cycle becomes [itex]\eta=1-T_C/T_H[/itex], i.e. the efficiency for the Carnot cycle.
But isn't it incorrect to consider the gas AND the "storage device" as one system, because there still is a heat flow (positive and negative) to the gas. Isn't it the heat flow to GAS that matters, when calculating the efficiency?!
And they've also derived that (the one above) formula for the efficiency of a stirling engine, in "Physics for Scientists and Engineers" By Fishbane, Gasiorowicz and Thornton, extended version.

Looking at your link, I see that the person who wrote it is working toward analyzing the stirling as a refridgeration device. In this configuration the engine is operated by an outside motor or engine to increase the temperature difference between the hot and cold end, and the cold end is then used to cool something. In this configuration the stirling engine is a heat pump and not really a stirling engine anymore. I think this is very likely the source of the confusion between you and your teacher.