I'd like to put forward an idea: just because alignment is a tangible thing in D&D (depending on the edition, most so in 3.X), doesn't mean everyone in-world necessarily has a really concrete notion of it, much less an opinion of how it works. How many people in the real world can name the four fundamental forces of physics, anyway? So I guess that most people that aren't spellcasters or paladins, and that don't have ranks in Knowledge (Arcana, Religion or The Planes) either have a very vague notion of it or even aren't aware of the concept at all. Furthermore, even for people who do know what alignment is and how it works, most won't be too interested in the philosophy of it, and just want to know how they put into words the result of their spell and move on with it. The whole argument about what "Good", "Evil" etc. really mean, or what alternate words they may use to refer to those concepts, and so on, is mostly for philosophy "wonks" who are really dedicated to this concept; most other people who apply the concept of alignment in a practical way probably just go along with whatever they learned from their mentor who taught them their character class, or from the local equivalent of Sunday school, you get the idea.

That said... my take on alternate interpretations of alignment is that people might assign different names and descriptions to the four cosmic forces of alignment ("Law", "Chaos", "Good" and "Evil") depending on how they interpret these forces, and especially where they sit in the alignment scheme themselves. Some ideas:

Good vs. Evil: The nomenclature of this alignment axis is very heavily biased toward the Good alignment; after all, extremely few people would agree with being called "evil", and even being Neutral means you're, well, not Good. On the other hand, there's the fact that "Good" people are arguably the ones that have the most moral investment, so to speak, in alignment; for Lawful, Chaotic and Evil people, the opposite alignment may be dumb, but it's not necessarily harmful or even monstrous, while that's certainly the case if you're a "Good" person thinking about the "Evil" alignment. This leads me to conclude that people of "Good" alignment would be very invested in the nomenclature of "Good" and "Evil", and unlikely to opt for alternate, less "biased" names.

Those of "Evil" alignment, on the other hand, are clearly the ones with the most reason to object to what the alignments are called. I think they'd probably say that "Good" and "Evil" are misnomers (or even harmful propaganda by Good-aligned races), and might refer to those instead as, say, "Weak" and "Strong", or "Passive" and "Driven" (or "Active"), depending on their exact interpretation.

Neutral (neither "Good" nor "Evil") folks may accept the nomenclature of "Good"/"Evil", but like I said, that's not very flattering to them, so perhaps they might adhere to the "Passive"/"Active" scheme, or even argue for something else, say, "Altruist" or "Communalist" for "Good" and "Egotist" or "Individualist" for "Evil", implying a dichotomy of focusing on other people vs. focusing on yourself, with the subtle implication that a balance between the two is ideal.

Law vs. Chaos: As has been alluded to in this thread, this nomenclature is also somewhat biased toward Law and against Chaos, although much less strongly than the other axis. I'd think it would be mainly "Chaotic" people that might object to it, since "chaotic" implies disorderly, random, meaningless, stuff like that. People of "Chaotic" alignment would probably use different words for "Lawful" and "Chaotic"; say, "Tyrannical" vs. "Free" if they're more confrontational, or "Rigid" vs. either "Loose" or "Free" if they're more amicable.

"Lawful" people would likely stick to the old "Lawful"/"Chaotic" scheme; while it's not perfect (Lawful people don't blindly follow anything that's written as law), they'd feel it gets the point across just fine, and is widely understood. However, those who are more chauvinistic about alignment might argue for the use of "Rational" vs. "Irrational" instead, since that's how "Lawful" people are likely to see this axis anyway.

People who are Neutral in the "Law"/"Chaos" axis probably wouldn't mind those alignments being called "Lawful" and "Chaotic"; again, they get the point across and are widely understood, and not being either doesn't sound that bad (well, "not lawful" sounds kinda like you're a criminal, but still). But, since these two names aren't quite that accurate, some Neutral folks might want to use other nomenclature instead; maybe "Rigid"/"Loose" (as mentioned above) if they want to sound more diplomatic, or even "Axiomatic"/"Anarchic" if they make a big deal of their neutrality and want to emphasize its value by making the other options sound extreme.