Question: Is there any term for the singing style of saying a lot of words very fast in a verse, as if you tried to squeeze what are clearly too many words in that verse? Is it just called rapping? It seems it could be something more specific than that.
Example: Jerusalem by Dan Bern
Specifically from about 1:05 when he says "... all the way to India..." and then the does it several more times
While there is not a term that I know of that specifically refers to singing "too fast" (apart from saying that someone is out of time), composers refer to how text is set as prosody. If someone clearly puts too many words into a given line, we would say that their prosody is poor or under-developed.
It is not rapping as rapping is not a sung style. The closest possible approximation would be recitative technique used by vocalists / composers in operatic and musical theater contexts in which performs sing-speak over repeated accompaniment.
Most of the rhythmic divisions used by the artist in the example are duple divisions, in other words, he is subdividing by powers of two in order to make the words fit in the appropriate time. He is not changing tempo or meter. His phrasing is sometimes stressed over the barline, and some syncopation results.
But by and large, when speaking about his delivery and use / treatment of the text, you would use "prosody" to describe what you mean.
Patter song is the term you are looking for.
The patter song is characterised by a moderately fast to very fast tempo with a rapid succession of rhythmic patterns in which each syllable of text corresponds to one note. (Wikipedia)
For example, here's a famous patter song from Gilbert and Sullivan's "Pirates of Penzance":
And here is an equally famous patter parody of said patter song:
