I'm no expert on paella, but I'm suspicious of definitions that depend on a particular tool or procedure. For years traditionalists like Marcella Hazan said that risotto was only possible if one followed a very strict set of procedures in an open pan with constant stirring. Now almost everyone who's tried it in a pressure cooker has kept doing so, including some prominent Italian chefs. We've come to our senses and defined risotto by the result, not by what the wizard is doing behind the curtain.
paulraphael, whereas I completelly agree with you in general, and agree in the case of risotto, I cannot agree in this specific case. I'm not a fundamentalist of paella (and there are quite a few around here), but rather I am defining paella in terms of its unique features, and these are: a completelly dry rice using short-grain rice whose grains do not join but feel separated, in a very thin layer in such a way that the bottom layer produces an intense maillard reaction (called "socarrat") and a kind of "oily" and very tasty top layer is formed. The rice must cook uniformly (and that's why a very thin layer is required), and each grain should have a slightly hard cender ("aldente"). Realize many paellas even here in Spain do not fit this definition, but it is what constitutes a real and good quality paella.
Getting these unique identifying features seems very hard without a pan that is quite wide and without a very controlled stock evaporation. In fact getting these results requires being extremelly careful with the rice/stock ratio and the heat control, which must be very strong during the first part of the cooking, then reduced, then increased again. Good paellas are one of the hardest recipes I know to get properly done.
Traditional paellas were cooked on fire on woods (called "sarmientos") which would also give smoke flavour to the rice. Modern paellas can be cooked on a variety of wide pans with differnt heat control, but other tools such as a pressure cooker or a rice cooker simply do not produce a cooking environment that can produce the characteristic features of paella. I am really open to new techniques and developments, but have not seen at the moment any tool or technique that can reproduce these identifying features.