"Teutschland" is in fact an old spelling of "Deutschland". The designation "deutsch" originates from the Old High German word "diutisc", which meant "belonging to the people". In short, the meaning was to differentiate speakers of Germanic languages like Franconian or Gothic from their neighbors who spoke Romance languages. Over the centuries and in different areas, several different spellings were used, some beginning with "deu-", some with "teu-", some with "doi-", some with "toi-" and so on.
Today, the correct spelling is "Deutschland". Other spellings are sometimes used in historical contexts or in yearning for supposedly better "olden times": "When William II was Emperor, such a thing wouldn't have happened!" Those "olden times" do not necessarily refer to the Third Reich or the German Empire, but can refer to about any period in German history.
Mostly though, other spellings of "Deutschland" are in my experience used today to mock somebody who supposedly yearns for those "olden times" or has backwater or far right views. Say for example, in a story there's a character who can't keep up with the changes in modern society, and doesn't intend to. Such a character could be saying things like "That's not the way things are done in Doitschland!!!".