  We perform a global fit to the experimental data of two-body charmed baryon
decays based on the topological diagrammatic approach (TDA) and take into
account the phase shifts between $S$- and $P$-wave amplitudes as inspired by
the recent BESIII measurement of the decay asymmetry in the decay
$\Lambda_c^+\to \Xi^0K^+$. The TDA has the advantage that it is more intuitive,
graphic and easier to implement model calculations. The measured branching
fractions and decay asymmetries are well accommodated in the TDA except for a
few modes, in particular, the predicted ${\cal B}(\Xi_c^0\to
\Xi^-\pi^+)=(2.83\pm0.10)\%$ is larger than its current value. The equivalence
of the TDA and the irreducible SU(3) approach (IRA) is established. We show
that the number of the minimum set of tensor invariants in the IRA and the
topological amplitudes in the TDA is the same and present their relations. The
predicted magnitudes of $S$- and $P$-wave amplitudes and their phase shifts are
presented for measured and yet-to-be-measured modes in both the TDA and IRA
which can be tested in the near future. Besides the decay $\Lambda_c^+\to \Xi^0
K^+$, there exist several modes which proceed only through $W$-exchange. In
particular, the observed channel $\Xi_c^0\to \Sigma^+ K^-$ should have phase
shifts similar to that in $\Lambda_c^+\to \Xi^0 K^+$ and its decay asymmetry is
predicted to be $-0.21\pm0.17$ which can be used to test our theoretical
framework. In contrast, the TDA leads to a large $\alpha$ of order $-0.93$ for
the decay $\Xi_c^+\to \Xi^0\pi^+$ even after the phase-shift effect is
incorporated in the fit.
