Let's see if this kicks up some discussion
Liking mathematics (or being a raging nerd) are not prerequisites for being an INTP
INTPs have dominant Introverted Thinking, which means that they approach life as a puzzle to be solved. They are drawn to exploring and expressing logic in all sorts of venues, not just mathematics. Other systems of thought could include philosophy, economics, films, video games, food, linguistics, even typology itself! INTPs just prefer it when things make sense to them, which creates a natural curiosity for systematic thinking and exploring logic.
Being "quiet and sensitive" does not automatically make you an IxFP
Dominant Introverted Feeling causes IxFPs to view the world in terms of the emotional needs expressed by every individual. Through this, they're able to assess the emotional (or "spiritual") congruence of whatever a person is doing (be it themselves or someone else), as well as consider how the actions of others might impede on that person's emotional needs. This results in a naturally empathic worldview, having compassion even for those that society would otherwise mock or discard.* Just because you find it hard to talk to strangers or express yourself emotionally does not make you an IxFP!
INxJs are not psychic
The extremely unbalanced ones might think they are, but they aren't. Dominant Introverted Intuition causes INJs to create new meaning by seeking strange or unique perspectives. Sometimes (but not always) this leads to predictions about the future, with an absolute certainty that's hard to explain. Sometimes it pans out as expected, often it doesn't. Healthy INJs are able to balance themselves with Extraverted Judgement, which provides a rational outlet for dealing with life and mesauring the validity of whatever strange thoughts they're thinking.
Just because you think you can't handle change doesn't make you a J
Unhealthy IPs can have a horrible time adapting to changing circumstances if they're too locked up inside themselves, unable to deal with the world using Extraverted Perception. Introverted Sensation causes them to adopt a dismissive attitude, which can sometimes resemble Extraverted Judgement. But without Extraverted Perception to guide them through the world, they don't have any way of dealing with situations that might otherwise seem chaotic or unusual. IPs need to learn how to adapt and negotiate with things outside of their control if they're going to live a happy life.
Post your own contributions!
* oh gosh I hope that description was accurate so I don't have to suffer anyone's wrath