No, you are not misunderstanding. If a triad is to work then each relationship has to develop. Otherwise the person coming in will always feel like an outsider. How are you supposed to feel when they say 'us' as in the two of them? In reality, I would rather have 'us' mean all three of us.
In my triad I make a conscious effort to say 'I' instead of 'we' when talking to my girlfriend. This actually has an impact on our thinking patterns since in this way I'm standing with her unconditionally. I don't need my husband to love her for me to love her. He doesn't need me to love her for he to love her. We all need to love each other individually in order to love together as three people.
I find the language for these concepts lacking. Whenever I write this stuff out I feel like I sound like an idiot. But, I will keep trying because apparently functional triads are unusual and anyone in a triad needs support.
It sounds like the couple you were involved with wasn't ready for this. I've learned that the couple has to be open to changing their relationship in order to form a triad. You can't just 'add another person' and expect your original couple dynamics to stay the same. Being in a triad forces us to keep changing, keep evolving the way we love each other. I'm very happy for it but it can be terribly frightening at times.
Good luck on your journey.