Yes
[to elaborate]
Green and White resemble each other (from a distance, if you squint) for a lot of the same reasons Red and Black resemble each other: in the simplified “Good Vs Evil” systems (like D&D), Black/Red is “Evil” and Green/White is “Good.” Green and White agree on things like “serving the community” and “the greater good.” They both believe in self-sacrifice, which is why they both hate self-serving, egocentric Black.
Where they differ, much like Red and Black, is in their relationship to Authority.
As you observed, White’s amicable relationship with Blue is indicative of White’s desire to enforce an agenda; they want to change the world. White’s agenda is nominally one of Peace and Harmony, but, unlike Green, White isn’t actually willing to take “No” for an answer.
White believes in Authority by Rule of Law; White loves Rules; White is an Empire Builder.
You see, things like Empire have certain requirements: for example, having an administrative ruling class necessitates a subservient labor class. Having “Rule of Law” necessitates things like “state monopoly on force” and “law enforcement.” And Green? Green spends waaay too much time hanging out with his buddy “all cops are bastards” Red to really have enthusiasm for “Crime & Punishment.”
In a room full of White soldiers, a man in a fancy hat makes a speech about sacrificing your life to secure the nation’s future. There is cheering.
In a room full of Green warriors, the exact same speech is met with “If I die in some field far from here, who exactly will protect my family? You?”
Not to imply that White doesn’t care about their family as much as Green does! They do! But “family” means something different to each of them.
White values the Family Name. White wants to have a bazillion babies and encourages them all to be doctors and lawyers and judges. White is a disciplinarian who believes in tough love. A good plan to make a White-aligned parent proud is to go forth and accomplish things.
Green values the Family Members. Green encourages their children to just be themselves, even if that’s not exactly in line with what society expects. Green prefers the carrot to the stick. One of the best ways to make a Green-aligned parent proud is to simply be there for your family.
The grim side of Green is that Nature is both brutal and savage, and Green doesn’t actually see a problem with that. Predation and Plague both have valuable roles to play in Nature, and Green respects that. Green is also skeptical of things like “private property” and has very little ethical issue with simply taking things (when in need).
So, yea. You hit the nail on the head there.