«زَيْدٌ» is a definite proper name but still takes tanwīn.
«أَحْمَدُ» is also a definite proper name but does not take tanwīn because it is a diptote.
Therefore, do not treat every occurrence of tanwīn as a marker of indefiniteness. Traditional grammar recognizes different kinds of tanwīn; here, focus on the noun's declension class.
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Munṣarif and Ghayr Munṣarif — Definitions, Causes and Parsing
«زَيْدٌ» is a definite proper name but still takes tanwīn.
«أَحْمَدُ» is also a definite proper name but does not take tanwīn because it is a diptote.
Therefore, do not treat every occurrence of tanwīn as a marker of indefiniteness. Traditional grammar recognizes different kinds of tanwīn; here, focus on the noun's declension class.