No, they are not the same thing. A compensated Poisson process is a Poisson process which is made into a Martingale by subtracting the mean from the process.

If the jump process has rate \lambda, couldnāt you just add a -\lambda into the drift term for the same jump-diffusion but using an uncompensated Poisson process?

yeah that is what I was thinking is the best solution at present. More variability would be nice but I am sure I can figure out how to get what I want of it.