There are indeed two different versions of the negative binomial distribution: (i) the distribution of the number (say $X$) of independent Bernoulli trials (with a given success probability $p$ in each trial) needed to have $k$ successes and (ii) the distribution of the number (say $Y$) of failures in an infinite series of such trials before the $k$th success. Obviously, $X=Y+k$. Each of these two "real-world" interpretations may have a certain appeal.
The existence of these two different (but closely and simply related) versions of the negative binomial distribution is perhaps unfortunate and confusing, but it is a fact. There is apparently nothing that can be done about it.