We consider the dynamics of systems driven by compound Poisson colored noise in the presence of inertia. We study the limit when the frictional relaxation time and the noise autocorrelation time both tend to zero. We show that the Itˆo and Marcus stochastic calculuses naturally arise depending on these two time scales, and an extra intermediate type occurs when the two time scales are comparable. This leads to three different limiting regimes which are supported by numerical simulations. Furthermore, we establish that when the resulting compound Poisson process tends to the Wiener process in the frequent jump limit the Itˆo and Marcus calculuses, respectively, tend to the classical Itˆo and Stratonovich calculuses for Gaussian white noise, and the crossover type calculus tends to a crossover between the Itˆo and Stratonovich calculuses. Our results would be very helpful for understanding relevant experiments when jump type noise is involved.