Au départ, je recherchais l'étymologie pour le nom anglais « clause », mais d'après ce qui suit (partie en gras), le changement clé s'est produit entre le latin et le français.
c. 1200, "a sentence, a brief statement, a short passage," from Old French clause "stipulation" (in a legal document), 12c., from Medieval Latin clausa "conclusion," used in the sense of classical Latin clausula "the end, a closing, termination," also "end of a sentence or a legal argument," from clausa, fem. noun from past participle of claudere "to close, to shut, to conclude" (see close (v.)). Grammatical sense is from c. 1300. Legal meaning "distinct condition, stipulation, or proviso" is recorded from late 14c. in English. The sense of "ending" seems to have fallen from the word between Latin and French.