Interesting idea, although I tend to be a little skeptical of equating IM elements or types with philosophical movements (though I may not be understanding you correctly). It's one thing to say empiricism is a predominantly Te philosophy. One could even construct a self-consistent "application" of typology in which Te is empiricism, and the other IM elements are other areas of philosophy, which is apparently what you have done. There's nothing wrong with this; I've constructed similar "reducts" of typology, though I consider it helpful to understand that that's what I'm doing...applying typology to one area, rather than saying that my interpretation equals the whole of essence of Jung/socionic typology.
it would be rather strange to me if in studying a phenomenon so closely related to the philosophy of thought we would never end up finding correlations between the two fields.

Specifically, empiricism, as I understand it, is a philosophical school that believes that all knowledge derives from the senses. While at first glance, it seems reasonable (since it's hard to even imagine intelligent existence of any sort without any inputs whatsoever), it's still a derived result, so I don't think anyone on the basis of being Te could arrive at that conclusion without at least first thinking quite a bit about it.

I'm not familiar with the various arguments for/against empiricism, but presumably the relevant questions are 1) Is there some sense in which mathematical or logical true is separate from what is known via the sense? 2) Are the claims of a mystic that he/she was communicated to in a non-sensory possible? 3) Could there be some sort of "instinctual" tendency that may count as a form of knowledge and would be inborn?

Ironically, if on the basis of Te, one could know the answers to those questions, that would be an argument against empiricism. In other words, if empiricism is true, and if merely being born a Te type causes one to know that it's true, then empiricism is false. Conversely, if a Te type merely thinks that's true but doesn't know it, then empiricism is also false (if thinking that something is true and not knowing means that one doesn't know it because it isn't true). (I suppose one could squeak by that and say that a Te type may believe it to be true without proving it, and it could still be true because the Te person's Te-ness did not make one know it even though it made one believe it and it's true, though believing something to be true that really is true is awfully close to knowledge.)
there are varying degrees of extremism at which the principle can be applied. i'm talking about a mild version where only an emphasis is placed on the demand for directly perceptible evidence. i don't see refutations of it's extremistic variety to be very useful to this discussion.

I understand labcoat as making references to metaphysics, not different schools of thought. "epistemic objectivity" is just a very specific way of saying "I believe knowledge comes from real things with studyable traits".
"real things" has more of an association with Pe. "studyable traits" gets pretty close to what i'm getting at. the issue is exactly that the two are hard to combine, because to jump from what we can study to what is real, we need to apply some sort of synthesis or abstraction.

Similarly, "ontological subjectivity" means "I believe what exists"
again i would connote this more with Pe. Pi is attentive to how things appear in a perspective-dependent way and holds skepticism to more perception-independent accounts of "what is real". however, this is also influenced by the Pi function being external (Si) among the person's valued functions, and possibly also by it's being Focal (in irrationals). in most type, the attitude is pretty ambivalent.

Irrational elements define one's ontological approach--"What exists, and what is it like?"
Rational elements define one's epistemological approach--"How can one know things?"
yes, but these things are constantly overlapping and not always capable of being isolated from one another. i do think you properly transcribe the terms.

Introtim elements are subjective:
Pi: "Things only exist as defined by my relationship with them; their properties are defined by how it influences me".
Ji: "One can only arrive at knowledge by following personal reasoning, be it discursive (T) or intuitive (F)."

Extratim elements are objective:
Pe: "The world is filled with objects that can be reduced to traits and analyzed."
Je: "I can obtain knowledge by studying objects and building predictable patterns in my head, be they arrived at through discourse or intuition."

I have a faint sense that I'm mixing a lot of meaninglessness in here, so you'll have to forgive my inexperience with metaphysics.
i'm not a big fan of the introtim/extrotim aspect of functions, because i suspect it doesn't have a technical definition of this kind. it's a qualitative connotation at best. attempts at finding the technical definition just have you end up with it's qualititative "feel" instead. i think this is what causes your self-professed sense of confusion.

I can illustrate it with an ESE...

The ESE believes that reality is wholly subjective, but builds knowledge by studying objects. They're basically blind to anything that they haven't concretely experienced (ontological subjectivism through ). However, the ESE builds knowledge by studying people, being an epistemological objectivist. "This signals implies this", "I can do that to effect the change I want". Ethics, being reasoned intuitively, can't really explain why it knows these things--it "just does"; while Sensation is pretty plainly obvious. You can't argue with an atom.

Also remember that your dominant function is always on. So, if you want to compare an ESE and an SEI, the SEI is living in a world of ontological subjectivity, the ESE is living in a world of epistemological objectivity.

And bam, we have Rationality/Irrationality as an input/output dichotomy, which is the most universal application of it, imo, and removes a lot of the taint that comes from the nurture aspects of it like "orderliness" (which is strictly purely orderliness-of-information-processing and not orderliness-of-character).
ethical functions are attentive to qualitative distinctions, so talking about epistemics and objectivity can be a little awkward in contexts like these.