Your explanation (user has n submissions with a score of at least n) fails to pass one of the Wikipedia examples. ([25, 8, 5, 3, 3] should result in an h-index of 3)
Yes, if there are 5 scores with at least 3 points, then there are 3 scores with at least 3 points. Of course he left out the fact that the h-index is the largest number for which the property he mentioned holds true, but that was obvious.