The current Watchtower organization has nothing in common with Judge Rutherford's teachings. They certainly have nothing in common with the early bible students and brother Russell.
So it is indeed correct to say that 'Jehovah's Witnesses' came into existence in 1931 since the name was adopted. Of course their roots date back to 1870s and to Russell. But they are as far removed from early bible students as they are from your friendly neighborhood Catholic.
The funny thing is that there wasn't even a governing body during Russell and Rutherd's time. Nathan Knorr held power until the mid seventies when the directors started to actually deal with matters (matters in relation to teachings and the brothers in general) and not simply corporate ones.