  We present a state-of-the-art reanalysis of experimental results on Efimov
resonances in the three-fermion system of $^6$Li. We discuss different
definitions of the 3-body parameter (3BP) for Efimov states, and adopt a
definition that excludes effects due to deviations from universal scaling for
low-lying states. We develop a finite-temperature model for the case of three
distinguishable fermions and apply it to the excited-state Efimov resonance to
obtain the most accurate determination to date of the 3BP in an atomic
three-body system. Our analysis of ground-state Efimov resonances in the same
system yields values for the three-body parameter that are consistent with the
excited-state result. Recent work has suggested that the reduced 3BP for atomic
systems is a near-universal quantity, almost independent of the particular atom
involved. However, the value of the 3BP obtained for $^6$Li is significantly
($\sim 20$%) different from that previously obtained from the excited-state
resonance in Cs. The difference between these values poses a challenge for
theory.
