There is of course no right or wrong way to visit the Cliffs of Moher. But I think the best way to visit this landmark depends on your travel party. For instance the viewpoint at Moher Tower at Hag’s Head is only accessible if you have a rental car. The official viewpoint can however be reached by bus/public transportation.
Also there is of course more infrastructure around the official viewpoint (café, souvenirs and visitor centre with information on the cliffs). The “Cliffs walking trail” is simply a mud trail next to the cliffs. And there are no handrailings either to prevent you from falling off said cliffs. So maybe if you have small children, the official viewpoint is a better option. This is also the case, if you are accompanied by someone who is not sure-footed or can’t climb over the field boundary walls.
The alternative at Hag’s Head however is a great option for people on a budget who do have a rental car. And also if you wish to just simply enjoy this nature landmark in peace. When we visited there were not many other tourists in this spot, we literally had the cliffs all to ourselves. I don’t know how it is during the summer though.