That illustration helps, but I think a better illustration might show what a box pleated kilt looks like when made with four yards, six yards, eight yards, etc.

On a traditional four yard box pleated kilt, the pleats generally look like the one simply called "box pleat" above. There is little to no overlap of the interior of the pleats.

Now, imagine the same pleating style but made with five yards, or six yards. There will be slightly more cloth. That means there will be slightly more pleats, and those pleats will be slightly more narrow. It also means that at least one side of the box pleats (typically the right side) will be deeper, to take in the extra cloth.

I, in fact, have a kilt like this. It's my newest kilt, box pleated, made from six yards of cloth.

It doesn't really look that much different from a four yard box pleated kilt from this angle, but if I were to take a picture from the "bottom up" the way the B&W line illustrations are, you'd see the pleats are much deeper on one side than the other. There are also about 12 pleats in that kilt, compared with the 8 or 9 pleats I typically get when I make a four yard box pleated kilt for myself.

If you continue this line of thinking for an eight yard kilt, the number of pleats, and the narrowness of those pleats, will be akin to what we are used to seeing in a modern day eight yard knife pleated kilt. But they will be box pleats. The left side of the pleat will be fairly shallow, while the right side will be deep, like a knife pleat. In fact, they look rather like knife pleats "turned in" on themselves. Which a lot of people, unfamiliar with the history of kilts and the various ways they are pleated, tend to think they are.

So the easiest answer to what the difference is between a box pleated kilt and a "military box pleated kilt" is that a military box pleated kilt is simply a box pleated kilt made from 8 yards of cloth. (And often not even that!)