To improve clarity and reduce book and BIP word counts let's change "secret seed" to "codex32-encoded master seed" and after introducing the concept, just "secret"?
"Seed" or "master seed" only for the secret's bytes data.
"Secret" for the string with index "s", "share" for non-"s" string.
Avoid adding adjectives that are redundant, oxymoronic, add jargon or blur this distinction.