Based on my own experiences, I will say this: I have one white parent and one Filipino parent, and I do not find the final quoted sentence to be true as a general rule here in the U.S. I'm always "half/half," "fil-am," etc. What's more I'd say, based on my own experiences and what I've observed of others, that the first paragraph is only true (when it's true) if the group in question is non-white. I mean, maybe that's indicative of white people thinking there's something wrong with being black/Filipino/whatever, but if you pass as white and have no connection to the "other" culture and someone learns you're 1/4 Japanese? People will start calling you Asian.
(Edit: I want to clarify the above paragraph to say that I was recalling a specific event in which a friend was trying to locate another friend and asked a store employee if they'd seen an "Asian" guy matching his clothing description. The point I was trying to make was that although that description would absolutely not be helpful in finding this individual, his Asianness remained a salient feature to the friend who knew he was part Asian. Although I think there is a question of the point at which a certain part of your ethnicity stops "counting" because it's so far back. Are his 1/8 Japanese children Asian? Their 1/16 children?)
I think there's a fine line between perpetuating the one-drop rule and acknowledging that that's simply how people are still regarded. People want people's races to be this OR that, not this AND that, and people are treated accordingly.