We consider two different semiclassical approximations for nonadiabatic quantum-mechanical correlation functions of the form Tr[(A) over cap e(e)(i (H) over cap)(t/h)(B) over cap e(g)(-i (H) over cap)(t/h)], where (H) over cap (g) not equal (H) over cap (e) represent the nuclear Hamiltonians of two different electronic states. The first approximation is based on direct linearization (DL) of the forward-backward (FB) action in the exact path integral expression for this correlation function. The second approximation is based on linearizing the FB action in an equivalent quantum expression for this correlation function, which is given in terms of the Meyer-Miller mapping Hamiltonian (MML). The two approximations have several features in common, namely: (1) They are given in terms of an integral over a classical-like phase space; (2) The relevant operators are replaced by their Wigner transforms; (3) The dynamics is purely classical and governed by a Hamiltonian that represents an average over H-g and H-e; (4) The fact that (H) over cap (g) not equal (H) over cap (e) gives rise to a phase factor of the form e(0)(iintegralt)(dtauU(tau)/h) dtau, where U = H-e - H-g. The main differences between the two approximations are: (1) The MML approximation involves an additional phase-space integral and Wigner transforms that correspond to the continuous variables representing the electronic degree of freedom; (2) The DL and MML approximations involve different averaged Hamiltonians, namely, (H) over cap (av) = ((H) over cap (g) + (H) over cap (e))/2 in the case of the DL approximation, as opposed to different relative weights of (H) over cap (g) and (H) over cap (e), which depend on the electronic degree of freedom, in the case of the MML approximation. The two approximations are tested within the framework of a nonradiative electronic relaxation (NRER) benchmark problem. Although the NRER rate constants are accurately reproduced by both methods, the DL approximation is consistently found to perform somewhat better. A discussion is provided of a feasible scheme for implementing those approximations in the case of anharmonic systems as well as the relationship to previous work.