You got the introverts mixed up.
Actually, no; for the introverts, Ji and Pi are what make them introverted, but they are accompanied by Je and Pe (in the auxiliary slot), which are what determine J/P. So it's IP: aux Pe; IJ: aux Je. Now, J and P match, but e/i are reversed. For introverts, it's it's mixed up because of J/P's definition as preferred extraverted function. But I know you don't believe this in favor of Jung's dominant function focus.
Another useful set of groupings are the Interaction Styles, but they are even more complicated as far as the type code.
EST/ENJ: In Charge
ESF/ENP: Get Things Going
IST/INJ: Chart the Course
ISF/INP: Behind the Scenes
These are basically the "social" temperaments. Like the ones you would get on the humour quizzes, and also, Social Styles, DISC, CPI, etc.
They are like this, because the social temperaments are basically "blind" to S/N. Social temperament is E/I and people/task (which comes to be called "informing/directing" in this model). S/N is tied to the other area of temperament; the conative (i.e. Keirsey, via Plato, Kant, Kretschmer).
So S/N end up denoting different "rules" for defining the people/task factor.
For N's, it's J/P, so it works like the earlier mentioned groups: EJ, EP, IJ, IP. (These are also known as the "sociability temperaments").
For S's, it's T/F, so the groups are ET, EF, IT, IF. These are groups I always thought made some sense, though they're hardly ever mentioned.
So Interaction styles are like a combination of these two letter groups. The result: T and J end up more "task focused", and F and P are more "people focused".