Well no, by their definition, if he consumed five or more drinks he wouldn't be a heavy drinker, he would have heavily drank. There is a meaningful distinction. The next sentence, as you'll have seen, is :

SAMHSA defines heavy alcohol use as binge drinking on five or more days in the past month.

And at that point, I totally agree. If someone is drinking more than this person, and on a regular basis, they would certainly seem to have a problem.

They are definitely not saying if you drink five beers one day you are now an alcoholic.