We exploit the decoherence of electrons due to magnetic impurities, studied via weak localization, to resolve a long-standing question concerning the classic Kondo systems of Fe impurities in the noble metals gold and silver: which Kondo-type model yields a realistic description of the relevant multiple bands, spin, and orbital degrees of freedom? Previous studies suggest a fully screened spin S Kondo model, but the value of S remained ambiguous. We perform density functional theory calculations that suggest S=3/2. We also compare previous and new measurements of both the resistivity and decoherence rate in quasi-one-dimensional wires to numerical renormalization group predictions for S=1/2, 1, and 3/2, finding excellent agreement for S=3/2.