The teacher interpreted the passage of the poem.
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The teacher interpreted the passage from the poem.

The Japanese, その詩の一節, means "a passage from the poem" (or one of the poem's passages.). Or, to not be too overly-influenced by the existing sentence, "a verse from the poem."

Most of the other translations use a definite article

But they could always be unlinked or changed. It did occur to me that "a" would apply to more aituations than "the", and now that I know that it's closer to the Japanese, I don't think there's a good reason to keep "the".

I wonder if "verse" would also work for the other translations.

Verse and poems, often go together. A "passage" seems more like something out of the Bible or prose.

I would say that a verse is a structural unit, whereas a passage or excerpt can consist of parts of multiple verses or a small part of a verse.

E.g. in Shakespeare's Sonnet 18, the following is both a verse and a passage/excerpt:

But thy eternal summer shall not fade,
Nor lose possession of that fair thou ow’st,
Nor shall death brag thou wand'rest in his shade,
When in eternal lines to Time thou grow'st.

While the following is just a passage/excerpt:

Nor shall death brag thou wand'rest in his shade,
When in eternal lines to Time thou grow'st.
So long as men can breathe, or eyes can see,
So long lives this, and this gives life to thee.

I agree.