Generic word for a verse, chorus, or bridge of a hymn or song

If you don't like section, the other term used by people in the music biz is part.

Masterclass:

The most popular songs tend to follow familiar patterns. While it’s possible to compose epic suites where no two parts are the same (looking at you, Dream Theater), most pop music sticks to traditional structures.

Tunedly

So, what are the structures often used in songwriting? First, let’s look at the parts that make up most songs... Chorus: The catchiest part of any song...

This can also be confused with the different usage of part which describes what each instrument is playing/vocalist is singing during each section, which is more common:

Masterclass:

ABAB Form. This form, called “binary structure” involves toggling back and forth between a verse section and a chorus section.

ABABCB Form. This is also known as “verse-chorus-verse-chorus-bridge-chorus” form. In this style of composition, the A section is the verse, the B section is the chorus, and the C section is the bridge.

Liveabout.com:

A bridge in songwriting is a section that differs melodically, rhythmically, and lyrically from the rest of the song.

Fretboard Anatomy

Perhaps the most basic aspect of music theory is giving names to the sections of a song.

Fender

Let’s put some labels for these sections and define the common things that occur in each section

So, according to the music industry, section is the way to label the segments of a song with different music.

Hymns are written in verses, and if there are lines to be repeated after each verse, these are called the refrain and are often printed in italics immediately after the first verse only. It's not necessary to print them again: the singers will know what to do.

The actual labels "verse" and "refrain" don't often appear in hymn books as they're not needed. More contemporary songs might use them though, and in this context the word chorus might be more appropriate than the rather old-fashioned refrain. (Or the even older-fashioned burthen/burden found in Shakespeare.)

If you need to describe different sections of the music, you can call them sections, as GArthurBrown says, and a linking passage (with different material in) might well be a bridge; without lyrics it might be marked instrumental.

What do you mean by "splitting on double line breaks"? Are you talking about double barlines?