Was the official name of the country the Weimar Republic
No. The official name was "Deutsches Reich", or "German Empire".
did people (both inside the borders, and outside) at the time it existed call those lands 'Weimar Republic'?
It seems not.
Or, at least, not in any significant numbers.
According to the Wikipedia page on the Weimar Republic:
"... *this name only became mainstream after 1933. Between 1919 and 1933 there was no single name for the new state that gained widespread acceptance, which is precisely why the old name Deutsches Reich remained even though hardly anyone used it during the Weimar period*."
The source for this is an article in Der Spiegel, titled Der Name des Feindes
Warum heißt die erste deutsche Demokratie eigentlich „Weimarer Republik?“. The article states:
" ... es während der gesamten Zeit der "Weimarer Republik" keinen Namen für den Staat gab, mit dem sich alle identifizieren konnten. Der offizielle Name, so entschied die Nationalversammlung, war "Deutsches Reich" - doch den benutzte eigentlich niemand, wie der Historiker Sebastian Ullrich aufzeigt: Auf der rechten Seite des politischen Spektrums wollte man dem demokratischen Staat den vermeintlichen Ehrentitel "Reich" nicht zugestehen. Bei der katholischen Zentrumspartei favorisierte man "Deutscher Volksstaat", bei der SPD "Republik"."
" ... during the entire "Weimar Republic" era there was no name for the state with which everyone could identify. The National Assembly decided the official name was "German Reich" - but nobody actually used it, as the historian Sebastian Ullrich points out: those on the right of the political spectrum did not want to associate the democratic state with the allegedly honourable title "Reich". The Catholic Centre Party favoured ""Deutscher Volksstaat"" ["German People's State"] with the SPD [Social Democratic Party of Germany] "[Deutsche] Republik" [German Republic]".
- my translation & my emphasis
According to the same Wikipedia article:
The first recorded mention of the term Republik von Weimar ("Republic of Weimar") came during a speech delivered by Adolf Hitler at a National Socialist German Worker's Party rally in Munich on 24 February 1929
going on to note:
Only during the 1930s did the term become mainstream, both within and outside Germany.