I have heard people referring to as a horizontal datum to models such as WGS84 and as a vertical datum to geoids. The former is primarily used for determining latitude and longitude but also the height (and thus X, Y Z can be calculated) above the ellipsoid while the latter is used for determining height above an equal-potential surface only.
I wonder why one would call "horizontal" a full 3D datum. The fact that you have a pure vertical datum (i.e. geoid) does not mean you have to call "horizontal" the remaining one.
Here are two articles where you can find the term:
http://geography.about.com/od/geographyintern/a/datums.htm
http://www.esri.com/news/arcuser/0703/geoid1of3.html
Best Answer
The term horizontal datum is used because it is more easily flattened into 2 dimensions and more useful for finding locations on a flat plane (compared to a vertical datum).
As in the ESRI post you referenced (image below), the Earth's surface is very uneven, so modeling this very difficult. The "ellipsoid" used in horizontal datums is close approximation of Earth's surface. While you can use a horizontal datum to determine height simply because it is a 3D model, it is not as useful as a geoid (used in vertical datums) which has some "physical" meaning because it approximates mean sea level fairly accurately. Therefore, heights are better represented or estimated with a geoid because they represent a physical estimate of mean sea level. Maybe someone else can explain it better..