As long as you can walk normally (using the whole foot not just the toe area) always hold your axe at its head with the blade pointing backwards.
This depends on the situation you are in. The text from Grivel seems to be a oversimplification. There is not just one technique for ascending and one for descending. There are two basic methods which are applicable for both ascent and descent (regarding how you hold the axe).
Moderate terrain / Walking on snow
This is likely the situation in question here, as in the question hiking and not mountaineering is mentioned. This includes both flat and steep terrain. The latter as long as you can walk with all points of the crampons in the snow (i.e. "normal" walking). Here you hold the axe at the head with the hand closer to the slope with the blade facing backwards. Here the axe is used to hold your balance and self arrest in case of a fall. The blade is facing backwards exactly for that: self arrest (the video in the question and this TGO question explain this well).
Steep terrain / Ascending on toe points
When the terrain gets too steep for the above technique one turns to face the snow and ascends plunging the toe pieces of the crampons into the snow. The axe is held at the shaft and arrested with the blade in the snow, so you can support your weight on it and use it to pull yourself upwards. Here a slip simply must not happen, the chances of being able to self arrest would anyway be slim. This is why you kind of "continuously" arrest with the blade facing forward. I guess this is what Grivel is talking about.