Unless my modular arithmetic is rusty these are actually the same thing (modulo a performance optimization), because of how you pick `p` and `q`, specifically the `p - 1 = nq` part.

An illustratory example:

1. Pick prime `p`. I picked `p = 7`.
2. Pick prime `q` so that `p - 1 = nq`. I picked: `q = 3`, `n = 2`
3. Pick `g` so that `g^q mod p = 1`. I picked `g = 2`.
4. Pick a random `k` between 0 and `q`. I picked `k = 2`.

`(2^2 mod p) mod q = 2^2 mod q = 1`

I've edited the text to remove the discrepancy, since I guess you're not the only person who will be comparing notes with Wikipedia :)