For the homogeneous crystal nucleation process in a soft-core colloid model, we identify optimal reaction coordinates from a set of novel order parameters based on the local structure within the nucleus, by employing transition path sampling techniques combined with a likelihood maximization of the committor function. We find that nucleation is governed by solid clusters that consist of an hcp core embedded within a cloud of surface particles that are highly correlated with their nearest neighbors but not ordered in a high-symmetry crystal structure. The results shed new light on the interpretation of the surface and volume terms in classical nucleation theory.