Only the lowest non-zero multi-pole moment is coordinate independent.
That's correct in spirit ... but all those lower-order zero-valued
moments are coordinate-independent also.
So I would have said:
The Mth moment is coordinate-independent if and only if
all lower-order moments vanish, where "lower order" includes
all orders from 0 to M-1 inclusive.
This follows immediately from expanding (r + r')^M via the binomial
theorem and checking whether it equals (r)^M.