From an individual cell, there are three movements available. But there are two kinds of cell - north pointing and south pointing - and the set of three directions you can move in from each is different.

From a north-pointing triangle, you can move south, northwest or northeast. From a south-pointing triangle, you can move north, southwest or southeast.

So across the grid as a whole, six movement directions exist.

I'm likewise unconvinced by that section of the article (which is interesting to read nonetheless!).

The benefit of hexes is that every neighbor is the same. There are no "special" kind of adjacencies, like with squares or triangles, where you must decide how to consider corners, whether to allow diagonals, etc. So hexes are the only ones that don't mess up distances nor require separate kinds of adjacencies.

Also: if tracing a line from the center point of the origin triangle, some of those paths require crossing another adjacent triangle (different from the "direction" triangle). This to me signals this triangular grid is wonky for wargames/games.