Actually, I am not sure that the viewpoints are diametrically opposing; rather as several have stated, 'IT DEPENDS' is the clear answer:
On depth, time, mix, surface interval, etc. There is no one right answer.
For example, if you use enriched air first, are you going to spend your entire (longer) NDL (60 minutes, at 70 feet, on 32%) at depth, or are you going to spend the same time at depth that you would have spent on air (40 minutes, at 70 feet, on 21%) - i.e. gas supply, NOT NDL, is the limiting factor. In the first case (longer dive time), you will have absorbed an equivalent amount of nitrogen, and the penalty you pay on the next - air - dive, is actually more substantial, because your air NDL is shorter anyway. However, a longer surface interval will offset this.
Personally, I would plan my deeper dive first. If the depth was within my enriched air MOD, I might choose to use that first, to reduce nitrogen loading. But, of course, that point is moot if I stay longer. If I stay a shorter time - the same time that I would have stayed on air, and then use air on the second shallower dive, the differential effect on NDL (air vs enriched air) may be small enough so that the effect on my dive time is trivial. Yes, I would have accumulated less nitrogen on the first dive, but I will accumulate more, on air, on the second.
If the dive is close to my MOD, I might choose air, then use nitrox on my second dive.
As Doppler said, and you decided to do, buy each other a beer. There is not RIGHT answer.
