The uncertainity principle does indicate indeterminism in quantum mechanics, in the sense that uncertainity in position measurement necessitates a corresponding uncertainity in momentum measurement. There is another aspect of indeterminacy in quantum mechanics in the context of measurements, which states that the exact outcome cannot be predicted. On the other hand, Chaos theory dictates the unpredictability of a classical system. There is a key difference between these two words - indeterminism and unpredictability. Indeterministic means what cannot be determined. Eg. In a quantum system, upon measurement, you cannot determine the eigenstate to which it collapses. With reference to Heisenberg's uncertainity, if you know the momentum of a system with some uncertainity, then automatically there will be a bound set on the uncertainity of the position of the system. (I think the word uncertainity in the uncertainity is a rather loose statement creating all kinds of misconceptions, the term Heisenberg's indeterminacy principle is more apt) That is if you take a large number of ensembles of the same state, then you will find that there is always a relation between the uncertainity of position and momentum which is not a characteristic of the measuring apparatus, but fundamentally of nature itself.
The unpredictability in classical systems arises due to chaotic behaviour of the systems. It is however different from the indeterminacy raised in quantum systems. Here the systems are deterministic here (not the case with quantum mechanics), you can exactly predict the future behaviour given the exact initial conditions. However, in practice, you get the initial condition only upto a finite degree of accuracy. This condition coupled with chaotic behaviour gives rise to unpredictability of the system. Note that if the exact initial conditions are known then by the laws of classical mechanics we can get the exact future evolution which is not the case with quantum mechanics, eg, while performing measurement, you don't know the exact future evolution of the system. Unpredictability in classical mechanics is thus more of a problem of determining the initial conditions of the system in question