I've been reading a couple of psychology books by Jonathan Haidt (The Happiness Hypothesis and The Righteous Mind) which I find very helpful in pointing out demonstrable psychological principles and clears up a lot of confusion for those who follow the Jungian side of things.
The interesting thing that the author notes is that modern psychologists have been treating emotion and cognition as distinct things, as a kind of dichotomy, and he determined that this is a severe category error. In other words, Jung had it right about F vs T!
Instead, emotion is a part of cognition just as reasoning is part of cognition. But while his categorical split aligns with Jung in some ways, it doesn't align in others. On the emotional side, he also places intuition and other automatic cognitive processes: all those things where we "see that" something is true. On the opposite side, he places all controlled processes such as conscious thought, all those things where we "reason why" something is true.
In some cases, these modes tend to align. You can give someone a moral dilemma and they both "see that" a particular resolution is true and can "reason why" that that resolution is true. But it's also possible to present dilemmas where these two different approaches yield different results. For instance, an experiment he wrote about was to ask someone to sign over their soul to the experimenter for $2. There was no mention of the Devil or anything religious like that, other than the word "soul" and giving it to the experimenter upon your death. The contract even said "This is just a psychology experiment and has no legal binding whatsoever." The experimenter said, "You can just tear up the contract after you sign it and keep the $2." The subject could even be an atheist (!) and they would usually absolutely refuse to sign over their soul. And those who refused could NOT explain why. They could "see that" they didn't want to sign over their soul, but they could not "reason why".
Similarly, you could show people a situation which violated some moral taboo (in this case, incest), but it added every possible condition that would make it OK: they were both adults, they both consented, they kept it secret (so they wouldn't hurt their family), they used birth control (to prevent a child with birth defects), it only happened once (and thus no fear of being discovered and thereby harming others by doing it habitually), and so on. All the "reasoning why" available indicated that incest was absolutely OK, but even with all of that, people would "see that" incest is "just wrong" but couldn't explain why, and all the moral reasoning available that made sure that no one was getting hurt and there were no victims of any sort couldn't justify it to them.
How does this relate to Ni, or functions in general? It doesn't map 100% to functions, but Ni is definitely on the "see that" side of cognitive processes. An Ni dom is "in tune" with that side and has trained oneself such that all their "see that" observations align with their "reasoning why" conscious understanding. Hence the connection to the unconscious that Jung observed about them.
In an INTJ, that training leads to intuitions about factual things. An INTJ will look at complicated math/science/logic problems and "see that" a particular solution is correct, intuitively avoiding all sorts of logical fallacies not by using logic but by knowing what all the fallacies "look like". Te plays the "reasoning why" role, here, where intuitions are compared with empirical evidence.
In an INFJ, that training leads to intuitions about emotional/people/social things, basically all the kinds of things that don't yield to strict logic and empirical evidence. There is much more "trusting one's gut" in an INFJ, because the Fe version of "reasoning why" doesn't provide a hard objective anchor to one's understanding: one can only see the effects that things have on people and work from there. So INFJs end up being extremely perceptive about people and certain kinds of complex systems, being able to intuitively "see that" certain solutions are true.
For completeness, I'll mention that INFJs are perfectly capable are intuiting logical/scientific/technical things and be very adept at it. The difference is that with Fe, they tend to learn these things more heuristically than analytically, so it will tend to be easier for INTJs to gain a logical/scientific intuition than for INFJs. Conversely, INTJs can develop a very adept social intuition that rivals that of an INFJ, but their approach to developing it is much more analytical, and people are much more easily heuristically evaluated (Fe) than analytically evaluated (Te), thus the INFJ has an advantage in such matters.
How does this map to the rest of the functions? I'm not sure, but my current guess is that the introverted functions (based on their overall tendency to be really stubborn and not lend themselves to verbalization) are part of the "see that" side of cognition in this model, while the extroverted functions are all on the "reason why" side of things. If this pattern holds, then for example it might mean that INTPs and INFPs aren't "intuitive because of Ne", but because Ti and Fi are particular kinds of intuitive/emotional approaches to cognition, and that ISFPs and ISTPs aren't "intuitive because of Ni", but because their dominant functions are introverted. It also implies that ISTJs and ISFJs rely very strongly on the intuitive "see that" approach, it's just a very "concrete" intuition as opposed to Ni's abstract intuition.
If I were to describe the functions such that these two models combine in a self-consistent way, I'd do so as follows
N = abstract reasoning
T = analytical reasoning
F = heuristic reasoning
S = concrete reasoning
Introversion (i) and extroversion (e) are NOT minor subsets of these, but essential overall categories, mapping directly to "see that" (i) and "reason why" (e). Or more aptly "unconscious reasoning" (i) that does not lend itself to words, and "conscious reasoning" (e) that very easily can be expressed with words.
Again, this is just speculation and playing with ideas on my part. I'm not trying to rewrite "official typology", whatever that means these days. I'm just trying to take different perspectives that arise from different kinds of observations, and synthesize them into a meaningful whole.