Well then why should we even bother making distinctions?
That's because the explanation is fuzzy thinking.
Intoversion is pretty basic. Across almost every test that measures the degree of introversion, there are the few notable traits. No one has all of them - these are simply bunches of common traits that make up introversion.
Here are the main traits within FFM (from Centacs) (MBTI down below) and their Introversion equivalent;
Enthusiasm - Holds down positive feelings
Sociability - Prefers working alone
Energy Mode - Prefers being still in one place
Taking Charge - Prefers being independent of others
Trust of others - Skepitcal of others
Tact - Speaks without regard for consequences.
In MBTI there are 5 sub traits that roll up into introversion - here they are with a loose definition of them (from CPP's STEP II Interpretive report)
Receiving - Reserved, Low-key, are introduced
Contained - Controlled, harder to know, private
Intimate - Seek Intimacy, One-on-one, find individuals
Reflective - Onlooker, Prefer space, Read and write
Quiet - Calm, Enjoy solitude, seek background.
House is strongly I in terms of FFM, but not so strongly I in terms of MBTI, although I would say far more I than E... a lot of this has to do with mapping to functions in MBTI.
Also, the thing to note is that these are meant to be universal traits - seeking attention at work to show off is only one small part of life. Doing it at work alone is a bad indicator - it means he exhausts himself for work because it is required. We all do that.
The problem with House is that he is a fictional character - his behaviour isn't consistent. He's also a drug addict, socially anxious and abused person. It is impossible to type him - even if he was a real person, he would not be considered a typable person.
However, his behaviour is very direct and linear. He is a problem solver - he comes in, solves a problem, goes home. An ENTP in that position would be nothing like him - he isn't an information gatherer, he isn't curious, not worried about the big picture, not open to others and not energetic. He carries through - to the point of breaking any rule he wants, using anything he wants and abusing whomever he wants. It is only the problem that prompts action. He has almost none of the extraverted traits - especially not the ones that are well supported biologically (emotions and seeking highs). Many of his particular quirks (his office being "his space" and needing it "his way") are decidedly introverted tendencies...
Seeing one extroverted tendency does not make him an extrovert - I've known quite a few INTJs and they are also very much attention seekers... in a specific narrow quality, typically in their area of expertise. In the very start of house, when asked why he had those team members, it was about "how to get things done" - he hired them based upon being cute, being a thief and... uhhh... damn, forgot the third (something to prove?). These people are pawns - all people are pawns. This is not an ENTP mindset... actually, not an E mindset at all.
As far as the writting thing goes - INTJs are gigantic writers. Journals, thoughts... The INTJ I started a company with wrote papers for fun... Non has journals going back 15 years - look at the blog now. INTJs here write about religion, about their lives... You'll note that this is even an introverted trait in MBTI. Plus, he's solving a problem - an INTJ will do just about anything to solve a problem.
That same INTJ has the nickname "Peacock" because he loves showing off his feathers... least, when it comes down to his interests.
Needing an audience is simply one small part of the show - a forced example to entertain. The way his life is - at home, outside of work, his hobbies...
I could just as easily say that House is a F because he has repeatedly risked his career to help patients (which is actually feasible since at the very start of the show it becomes clear that if he meets a patient - considers them human - he ends up respecting them).