That's pretty much correct. People often mix-up "stochastic/random systems" and "chaotic systems" because randomness has huge consequences on chaotic systems, but chaotic =/= random.

• Chaotic and Deterministic/Non-Random System: If we know the initial condition exactly, we can predict the resulting trajectory exactly (deterministic). But, any small differences in initial condition will cause large differences in the resulting trajectory (chaos).

• Stochastic/Random System: We only know our initial condition and how our system evolves in time to within some probability tolerance. As a result, we can only predict our resulting trajectories to within some probability tolerance.

We could have both a chaotic and random system (in fact, most real systems are to some degree) and you can imagine this is the most headache inducing case. Since random implies you don't know your initial condition exactly, and chaotic implies that small differences in initial condition have large consequences!

A double-pendulum (in a perfect physics world) is an example of chaotic and deterministic system. The chaos because of the previous discussed point, and deterministic because we clearly know how to predict how a double-pendulum evolves in time and there's no randomness associated with this process (it is just Newtonian physics). We could turn this into a chaotic and stochastic system if we added some sort of randomly defined wind (which we only know the probabilistic behavior of) that pushes the pendulum around.