  It is customary to use a precessing convention, based on Newtonian orbital
angular momentum ${\bf L}_{\rm N}$, to model inspiral gravitational waves from
generic spinning compact binaries. A key feature of such a precessing
convention is its ability to remove all spin precession induced modulations
from the orbital phase evolution. However, this convention usually employs a
post-Newtonian (PN) accurate precessional equation, appropriate for the PN
accurate orbital angular momentum ${\bf L}$, to evolve the ${\bf L}_{\rm
N}$-based precessing source frame. This motivated us to develop inspiral
waveforms for spinning compact binaries in a precessing convention that
explicitly use ${\bf L}$ to describe the binary orbits. Our approach introduces
certain additional 3PN order terms in the orbital phase and frequency evolution
equations with respect to the usual ${\bf L}_{\rm N}$-based implementation of
the precessing convention. The implications of these additional terms are
explored by computing the match between inspiral waveforms that employ ${\bf
L}$ and ${\bf L}_{\rm N}$-based precessing conventions. We found that the match
estimates are smaller than the optimal value, namely 0.97, for a non-negligible
fraction of unequal mass spinning compact binaries.
