PoppaGator wrote:
"[...] I still don't know what word is used for that most basic part of a song in contexts where the word "verse" is used to describe the obscure and often forgotten introduction."

This is a bit of a sore point with me, as someone who will sing verses, even though they are often somewhat banal. The verses were not meant to be cut and the songs were meant to be sung the way they were written. Sure, many jazz musicians have played and sung wonderful versions, but the songs (and the musicals) are works of art in their own right and deserve to be sung as they were meant to be sung. Not that they always have to be sung this way, of course.

(Popular music of this era, in English and German, is a big part of my repertoire.)

Often, there is more than one verse (and sometimes choruses with different words). Fake books often leave out the verses and jazz musicians often don't play them. Even sheet music and song books of composers and lyricists such as Cole Porter, Jerome Kern, Noel Coward, the Gershwins, Irving Berlin, etc., don't always include the complete lyrics. For some of them, there are books containing just the lyrics. Often the lyrics differ from what was actually sung in the theater or in the movie. One example is Dubin and Warren's "Shuffle Off to Buffalo" from the film "52nd Street", which was fairly elaborate production number (directed by Busby Berkeley), though not one of the more spectacular ones (no waterfalls). This is one reason why I don't like fakebooks (even the legal ones).

I will go to some trouble to find the original, correct and complete lyrics of popular songs.

One reason verses are often left out may be that they were, as I said, often somewhat banal. Another was probably the limit of about 3 min. for one side of a 78 record. On recordings of, say, Fats Waller or Teddy Wilson, there might be two or three instrumental solos, so the song couldn't be sung complete. If I'm just playing and not singing, I will sometimes leave out the verse.

Incidentally, Kurt Weill's verses are seldom if ever banal.