The situation became a little more complicated than your normal parent/kid struggles, though. My parents divorced when I was 7 and afterwards I was my mother's entire support network - she had no friends, no family she could rely on, and it took her years and years to find a job where people could appreciate her tremendous strengths and talents instead of feeling judged and steamrollered by her which finally allowed her to make friends with colleagues.
So she put me in a position where I had to be her strength. And she is my mother and I love her and of COURSE I wanted to support her in whichever way sne needed, but a pre-teen and young teen isn't cut out to have her mother cry on her shoulder because of depression/being unappreciated at her job/not being able to pay rent or decent food. I did the best I could, but I didn't have any support network of my own either. If she'd just let me feel miserable by myself and stayed out of my head, I could've been strong for her. If she'd just been happy and taken care of and supported, I could've been open to her. But she leaned on me beyond any point of skill I had and then she poked at me because I refused to share my emotions and THEN she kept telling me those emotions were wrong and a kid should be happy, dammit! *sad*
I think your response to INTJ daughter was lovely and just right and I would've given my right hand for an interaction like this. Even with the poking fun on FB afterwards. :p
With my mother, I'm definitely allergic to misreads. Mostly because she wouldn't accept any correction. Sure, she gets bellyaches when she's nervous, so if I complain about a bellyache that's a great starting point in the guessing game. But when I say "No, not nervous, just cramps" I need people to accept that. Instead, she'd insist that I was TOO nervous and when you're nervous you need to walk it off while talking about it. (Whereas I just wanted to go to bed and sleep.) And the other way around, when I felt nauseous (a sign of nerves in me, a sign of illness in her), she'd put me to bed and not allow me to walk it off/talk about it. So it's not so much the mislabeling that gets me as the resulting 'so you have to do THIS' that interferes with self-care and leaves me both miserable for longer AND feeling misunderstood.
As a grown-up, what works for me is "Hey honey, I get the feeling you're not feeling too good. Anything I can do to help? Wanna talk about it? Want me to cook dinner?" (Any offer of cooking dinner always works like a charm - it helps me take care of things by proxy by feeding my body real food, it helps me be less stressed because there's one bit of responsibility I can let go of, and it gives me 20 minutes I can spend on delving deeper into whatever's bugging me because I don't have to entertain you because you're entertaining yourself by making dinner. My husband tends to say "Looks like you're not feeling too good - how about we skip dinner so you don't have to worry about that!" and that doesn't help at all. Poor soul.)