  We revisit the problem of well-defining rotation numbers for discrete random
dynamical systems on the circle. We show that, contrasting with deterministic
systems, the topological (i.e. based on Poincar\'{e} lifts) approach does
depend on the choice of lifts (e.g. continuously for nonatomic randomness).
Furthermore, the winding orbit rotation number does not agree with the
topological rotation number. Existence and conversion formulae between these
distinct numbers are presented. Finally, we prove a sampling in time theorem
which recover the rotation number of continuous Stratonovich stochastic
dynamical systems on $S^1$ out of its time discretisation of the flow.
