  Physical observables, such as the scattering phase shifts and the binding
energies, calculated from the non-local HAL QCD potential do not depend on the
sink operators used to define the potential. This is called the scheme
independence of the HAL QCD method. In practical applications, the derivative
expansion of the non-local potential is employed, so that physical observables
may receive some scheme dependence at given order of the expansion. In this
paper, we compare the $I=2$ $\pi\pi$ scattering phase shifts obtained in the
point-sink scheme (the standard scheme in the HAL QCD method) and the
smeared-sink scheme (the LapH smearing newly introduced in the HAL QCD method).
Although potentials in different schemes have different forms as expected, we
find that, for reasonably small smearing size, the resultant scattering phase
shifts agree with each other if the next-to-leading order (NLO) term is taken
into account. We also find that the HAL QCD potential in the point-sink scheme
has negligible NLO term for wide range of energies, which implies a good
convergence of the derivative expansion in this case, while the potential in
the smeared-sink scheme has non-negligible NLO contribution. Implication of
this observation to the future studies of resonance channels (such as the $I=0$
and $1$ $\pi\pi$ scatterings) with smeared all-to-all propagators is briefly
discussed. All computations in this paper have been performed at the lattice
spacing $a\simeq 0.12$ fm ($1/a \simeq 1.6$ GeV) on a $16^3\times 32$ lattice
with the pion mass $m_\pi\simeq 870$ MeV.
