If basis set is complete (finitely large basis set), R length and R velocity will be the exactly the same. However, since R length converges with respect to basis set size more rapidly than R velocity, therefore R length is more recommended to use in practice. If you use a larger basis set, perhaps the difference will be notably smaller.

"origin-independent" denotes that the result keeps exactly unchanged when you translating the system, which is equivalent to displacement of origin.

In most of the calculations I have performed the values of R length and R velocity are quite similiar or nearly identical, but I have found a case where there is a transition (the first one) with a very different value for both magnitudes. Have you any idea about this phenomena?

Thanks in advance