New alignment system (D&D any, PEACH)

[Xuldarinar]

I'm afraid that your interpretation of the rules is the incorrect one. If you continue to read on in the very same section of the SRD, you'll find the following quote:

"These paladin variants aren't meant to be unique classes in and of themselves, but rather alignment-based variations of the paladin. They have the same Hit Die, skill points per level, weapon and armor proficiencies, and spells per day as the standard paladin. Their class skill lists are nearly identical, with exceptions noted below. Their spellcasting functions identically to that of the standard paladin (though their spell lists are somewhat different). When a class feature has the same name as a paladin class feature, it functions the same as the one described for the standard paladin." (emphasis added).

This strongly suggests that the paladin variants fall under the "variety of paths" category rather than the "wholly separate class" category. Ergo, multiclassing between them is barred.

"...In cases where a single class offers a variety of paths (such as the totem barbarian or the monk fighting styles), the easiest solution is simply to bar multiclassing between different versions of the same class (just as a character can't multi class between different versions of specialist wizard)..."

In this case it still doesn't say they are barred. It says the easiest solution for handling them is barring them. By RAW its to the DM's discretion.

You do however make a good case, and if a DM followed what they suggested would be the easiest thing to do then it would likely be barred.