Definitely. People will name tunings any way they want. I'm not sure what your reference page is from. I agree that there are obvious errors if the goal is to name an actual chord. But that may not always be the point. I am also not sure whether some of these "inconsistencies" are from actual usage, or errors made by your source. But it is much ado about nothing, IMO.Maybe this question is overly pedantic ...
In the first place - the name of a tuning does not tell us what the tuning is. There is not a one-to-one correspondence between actual tunings and names - the same name may refer to different tunings, and the same tuning may be referred to by different names. How many ways can one implement a C6 tuning, for example? Plenty. There are dozens if not hundreds of different steel guitar tunings. The only way to know exactly which tuning one is talking about is to spell it out.
In the second place - is it really necessary to name a tuning by accurately describing which Western-notation chord is made from it? And if there are multiples, which one? Or which ones? Example:
For me, D/G is more descriptive of the use of the tuning because Gmaj9 doesn't come close to fully describing how I would use the tuning - it completely misses the D-root usage. But if you want to be pedantic, D6/Gmaj9 pretty well covers it because the top 5 strings clearly form a D6 chord and the bottom 5 strings clearly form a Gmaj9 chord. To me, the only utility of the labeling is the make clear various functions of the tuning. But I still want to know exactly which notes are in the tuning. As far as I'm concerned, anything else is ambiguous.My 6 string tuning is G B D F# A D. I call it D/G, pronounced “D over Gâ€. I think that’s better than the chord name G Major 9, don’t you?
As far as labeling of tunings from low-to-high, or vice versa - people will write tunings any way they want. They will write string gauges any way they want. I think it would help if people stated which way they're doing it - there clearly is no generally agreed upon "standard". Personally, I prefer low-to-high because I think in intervals from low-to-high. But I don't think it matters a whit which way one diagrams it, as long as the writer states what they're doing so there is no confusion.
On the other hand, it would be a rare tuning indeed for which a reasonably experienced player couldn't quickly infer the correct order. However, for the benefit of less experienced players, I think it is important to state the order explicitly.