In practical terms, is the difference only the weight term added to the fraction (probability) of balls of color c?
Hypergeometric distribution describes the number of observed white balls $k$ out of $n$ draws without replacement from the urn containing $K$ white balls and $N-K$ black balls.
Multivariate hypergeometric distribution is a multivariate distribution that describes the number of observed balls in different colors $k_i$ out of $n$ draws without replacement from the urn containing $K_i$ balls of each color, where the total size of population is $\sum_i K_i = N$ balls.
Noncentral hypergeometric distributions (there are two Wallenius' and Fisher's) is a generalization of (univariate) hypergeometric distribution, where white balls and black balls have unequal "weights", what influences the probability of being drawn. The two noncentral distributions differ from each other and differ from the hypergeometric distribution, so I'd encourage you to read about them in detail since there are more differences.