  Recently, two independent calculations have been presented of finite-mass
("self-force") effects on the orbit of a point mass around a Schwarzschild
black hole. While both computations are based on the standard mode-sum method,
they differ in several technical aspects, which makes comparison between their
results difficult--but also interesting. Barack and Sago [Phys. Rev. D {\bf
75}, 064021 (2007)] invoke the notion of a self-accelerated motion in a
background spacetime, and perform a direct calculation of the local self force
in the Lorenz gauge (using numerical evolution of the perturbation equations in
the time domain); Detweiler [Phys. Rev. D {\bf 77}, 124026 (2008)] describes
the motion in terms a geodesic orbit of a (smooth) perturbed spacetime, and
calculates the metric perturbation in the Regge--Wheeler gauge (using
frequency-domain numerical analysis). Here we establish a formal correspondence
between the two analyses, and demonstrate the consistency of their numerical
results. Specifically, we compare the value of the conservative $O(\mu)$ shift
in $u^t$ (where $\mu$ is the particle's mass and $u^t$ is the Schwarzschild $t$
component of the particle's four-velocity), suitably mapped between the two
orbital descriptions and adjusted for gauge. We find that the two analyses
yield the same value for this shift within mere fractional differences of $\sim
10^{-5}$--$10^{-7}$ (depending on the orbital radius)--comparable with the
estimated numerical error.
