I know the German name for their country is Deutschland meaning essentially Land of the People. And presumably the english word is derived from the Roman province. But other parts of Europe now uses some variant derived from the name of a single Germanic tribe (ie. French is Allemagne, Spanish is Allemania).
How and Why did English retain the Roman name while other more Romanic languages changed? When was the name first used for a specific area approximately corresponding to modern Germany?
Let me give you a breakdown:
Deutsch: Many language use a variation of this, not only German itself in Deutschland, but also Dutch Duitsland, Danish, Norwegian and Swedish Tyskland, Icelandic Þýskaland, Japanese ドイツ doitsu (emulating Dutch duits), and Mandarin 德國 déguó or 德意志 déyìzhì. As you already know, this derived from Old High German diutisk meaning 'popular'. In contrast to the official Latin language of the time in the Carolingian Empire and its successors, the German language was called the language of the people. That's where this comes from. Teuton, by the way, comes from a distantly related Celtic word *teutā, meaning 'people'
Germania: English Germany, Italian Germania, Hindu जर्मनी Jarmanī, Russian Германия Germaniya, and Swahili Ujerumani are a few examples of names derived from the Latin therm Germania, already metioned by Tacitus and Caesar. They probably took the word from the Gauls, who referred to the peoples beyond the Rhine as Germani. Whether this word originally meant 'neighbour', 'battle cry', 'spear', or 'noisy' is unclear. This is probably the first name for the German region as a whole (although the exact borders varied). This name spread through Latin and later English.
Alemanni: Variants of this include above mentioned French Allemagne and Spanish Allemania, as well as Arabic ألمانيا 'Almānyā, Turkish Almanya, Kurdish Elmaniya or Welsh Yr Almaen and obsolete English Almain, borrowed from French. The Alemanni, which according to Asinius Quadratus means 'all men', were a group of tribes often identified with the Suebi or Swabians. Swabia is the southwesternmost edge of modern Germany and in the Middle Ages for people from Romance-speaking areas the place where the people started speaking weird aka German. This would be the reason, why the term for this particular reason was extended to the entirety of Germany.
Saxon: Similarly, in Finnic languages and Romani, the name for Germany is derived from the tribe of the Saxons or the region of Saxony, being close to the Northeastern edge. There is Finnish Saksa, Estonian Saksamaa, Romani Ssassitko temm or Livonian Saksāmō
Nemets: Derived from this Proto-Slavic root meaning 'mute' aka not able to speak Slavic, are many Slavic names for Germany, for example Czech Německo, Polish Niemcy, Serbo-Croatian Немачка Njemačka, as well as Arabic نمسا nímsā and Ottoman Turkish نمچه Nemçe, meaning Austria.
Vagoths: Baltic languages use a term possibly derived from a tribe called Vagoths, who migrated to Germany from Gotland, which is an island in the Baltic Sea, or from a word meaning 'speak'. Lithuanian Vokietija and Latvian Vācija belong to this group.
As this diversity shows, it is clear, that for a long time Germany was not regarded as a unified region. Notion of a German Unification came about with nationalism, but before that, the only connection were linguistic and sometimes central authorities with very limited control, like the Holy Roman Emperors. Without a central authority, as was the case in the Early Middle Ages as well as Roman times, the name for Germans was either linguistically motivated (speaker of the popular language, non-Slavic speaker) or the closest tribe/region, which then later came to mean the entirety of Germany.