>>Etymology of the word “Dutch”, from http://www.etymonline.com (great resource, btw)
Dutch - mid-14c., used first of Germans, after c.1600 of Hollanders, from M.Du. duutsch, from O.H.G. duit-isc, corresponding to O.E. þeodisc “belonging to the people,” used especially of the common language of Germanic people, from þeod “people, race, nation,” from P.Gmc. *theudo (see Teutonic). As a language name, first recorded as L. theodice, 786 C.E. in correspondence between Charlemagne’s court and the Pope, in reference to a synodical conference in Mercia; thus it refers to Old English. First reference to the German language (as opposed to a Germanic one) is two years later. The M.E. sense survives in Pennsylvania Dutch, who came from Germany. Since 1608, Dutch has been “an epithet of inferiority” – Dutch treat (1887), Dutch uncle (1838), etc. – probably exceeded in such usage only by Indian and Irish. Dutch elm disease (1927) so called because it was first discovered in Holland (caused by fungus Ceratocystis ulmi).
>>and etymology of the word “Teutonic”:
Teutonic - 1605, “of or pertaining to the Germanic languages and to peoples or tribes who speak or spoke them,” from L. Teutonicus, from Teutones, from O.H.G. diot “people” (see Dutch), from *teuta, the common PIE word for “people” (cf. Lith. tauto, Osc. touto, O.Ir. tuath, Goth. þiuda, O.E. þeod). Used in anthropology to avoid the modern political association of German; in this broader sense Fr. uses germanique, Ger. germanisch, since neither uses its form of German for the narrower national meaning (Fr. allemand, Ger. deutsch). The Teutonic Knights (founded c.1191) were a military order of Ger. knights formed for service in the Holy Land, later crusading in Prussia and Lithuania.
>> If you’re wondering what PIE means, as in: "from *teuta, the common PIE word for “people” (cf. Lith. tauto, Osc. touto, O.Ir. tuath, Goth. þiuda, O.E. þeod). ", I believe it stands for Proto-Indo-European.
I guess this suggests that the germans have co-opted the proto-indo-europan word for “people” to use on themselves. 