I intended the comment about multiplicity to be an exception to the first line about quadratic convergence. You are correct that this is less than clear.  Your suggested phrasing is fine.  Durand-Kerner is Newton's method simultaneously on a vector of root guesses, with the derivative of the polynomial approximated by the derivative of what the polynomial would be if its roots were the current set of guesses.  This permits a lot of redundant computation to be eliminated,  and the algorithm parallelized.  If f(x) is a 4th degree polynomial and p,q,r,s are root approximations, the approximated polynomial is (x-p)(x-q)(x-r)(x-s) and its derivative at p is (p-q)(p-r)(p-s) so a Durand-Kerner iteration for p would be p = p - f(x)/((p-q)(p-r)(p-s)) and iterations of identical form for q, r, and s follow immediately.  You are correct this looks a lot like the example given in the Jacoby's method article.