  We prove an abstract result establishing that one can obtain the convergence
of Rare Events Point Processes counting the number of orbital visits to a
sequence of shrinking target sets from the convergence of corresponding point
processes for some induced system and matching shadowing shrinking sets inside
the base of the inducing scheme. We apply this result to prove a dichotomy for
two classes of non-uniformly hyperbolic interval maps: Misiurewicz quadratic
maps and doubly intermittent maps. The dichotomy holds in the sense that the
shrinking target sets may accumulate in any individual point $\zeta$ chosen in
the phase space and then one either obtains a limiting homogeneous Poisson
process at every non-periodic point $\zeta$ or a limiting compound Poisson
process with geometric multiplicity distribution at every periodic point. We
also highlight the reconstruction performed in order to recover the
multiplicity distribution for a periodic orbit sitting outside the base of the
induced map.
