Using the 165.9-keV M1 transition in (139)La as a calibrant, we have determined the K-shell internal conversion coefficients (ICCs) for the 127.5-keV E3 transition in (134)Cs and the 661.7-keV M4 transition in (137)Ba to 0.6% precision. Unlike previous less-precise measurements of these ICCs, which differed significantly from calculated values, our results agree well with calculations that include the atomic K-shell vacancy and disagree with those that exclude it. These results, which are consistent with what we also found for (193)Ir, confirm our earlier conclusions but for different energies, multipolarities and atomic numbers.