I certainly think the MBTI as a questionaire and the encoding of types into 4-letter codes is badly designed. But I think the interpretation of underlying theory based on cognitive functions and functional stacks is both elegant and compelling.
Taking your description naively, such a person would likely be an INTJ. Which has a functional stack of (in order of most to least used thinking processes): Introverted Intuition - Extraverted Thinking - Introverted Feeling - Extraverted Sensation.
I realise that doesn't make much sense if you are used to applying the 4-letter code to persons directly. But as I mentioned above, I don't think that is a good way of doing things. It makes more sense if you take the 4 letter codes as simply being an opaque shorthand encoding for functional stacks (with the individual letters in the code not having meaning on their own, only indirectly through their correlation to a functional stack). E, I, N, S, T, F, P, J then become properties of thinking processes and an "extraverted person" is simply someone who primarily tends towards using one of the four extraverted thinking processes (Fe, Te, Ne, Se). They may or may not present like the everyday understanding of how an extravert presents (e.g. sociable and chatty). Typically people who have a "Extraverted Feeling" (Fe) dominant function closely fit this stereotype, but other types extraverts often don't.
Ok, that makes sense, but you should probably clarify that you have a nonstandard view of MBTI when you say that people can't be the personality type they describe themselves as.
If someone did the test and got INTP, then I think it is fair for them to describe themselves as INTP. You might say that the test was wrong, but as I said that isn't how most see MBTI.
It's actually not that non-standard. It's how the majority of people who are into MBTI / jungian type view things (where I've linked MBTI to Kahneman's Fast/Slow thinking is non-standard, but a cognitive functions + functional stack view is not).
It's also what the more advanced "Step III" version of the official MBTI test is based on. But that that assessment isn't just a simple questionaire: it can be only be conducted by a trained assessor and the cost of training course runs to thousands of pounds, so most people don't have experience of it, and it ends up being the simpler naive model that most people learn about.
Taking your description naively, such a person would likely be an INTJ. Which has a functional stack of (in order of most to least used thinking processes): Introverted Intuition - Extraverted Thinking - Introverted Feeling - Extraverted Sensation.
I realise that doesn't make much sense if you are used to applying the 4-letter code to persons directly. But as I mentioned above, I don't think that is a good way of doing things. It makes more sense if you take the 4 letter codes as simply being an opaque shorthand encoding for functional stacks (with the individual letters in the code not having meaning on their own, only indirectly through their correlation to a functional stack). E, I, N, S, T, F, P, J then become properties of thinking processes and an "extraverted person" is simply someone who primarily tends towards using one of the four extraverted thinking processes (Fe, Te, Ne, Se). They may or may not present like the everyday understanding of how an extravert presents (e.g. sociable and chatty). Typically people who have a "Extraverted Feeling" (Fe) dominant function closely fit this stereotype, but other types extraverts often don't.