Borrowed from the literary tradition, I'd like to put forward some new words for your perusal that might help explore the differences of goals that exist under the catch-all word immersion. There may be others that would help too, but I think these three are important.
Catharis: Yes, I know you know this word, but do you know what means in the context of literary theory? Catharsis (which was introduced by Aristotle in The Poetics and means either "purgation" or "purification" in Greek) is the emotional effect a tragic drama has on its audience. The audience of a tragic drama would experience an overwhelming feeling of exaltation or relief following the drama because either they formed a vicarious identification with the hero which cleansed the emotions as if they have themselves had undergone the trauma of the story, or because the audience becomes so engrossed in the emotions for the hero that they are removed from the context of their own lives and return refreshed and renewed back to themselves following the drama.
Kairosis: is associated with the epic novel (association with the Greek meaning "the right time", and represents the feeling of integration experienced by the audience with the protagonist. It is associated specifically with the moments of moral and psychologicical transitioning of the character in important, dramatically impacting moments. It is interesting to note that Kairosis is often achieved by challenging unique dynamic characters with typical, everyman dilemmas and emotionally engaging in the moments of change.
Kenosis: is associated with lyric poetry, and represents the audience's abandonment of the ego manifestation in favour of the immediate emotional body and sensory manipulation of the poetic. It comes from the Greek word for "emptiness" and is used to achieve a feeling of timelessness or transcendence.
(If you don't care about the words in their application in literary theory, you can skip this indented part.
***ETA: There's more discussion after the indented part. Pick up the post again in the paragraph starting with: "So, what the heck am I talking about?"**
When I look at these terms, I make some adjustments on them to compensate for the differences in the method and process of the act of roleplaying:
Where we in a widely literate, educated and media saturated environment have specified, culturally driven, inherited understandings of drama, and in a world where the lines between the novel the drama and lyric poetry have been distorted, deconstructed and blurred, it seems to me that goals may not cleanly align by the form but can still maintain similar extant resonance to the emotional outreach of the audience.
Where we, as roleplayers, serve as both the authors and audiences of our own characters, inside a dynamic, living drama rather than a static text, we can elect to chase the fulfillment of multiple goals at once.)
So, what the heck am I talking about? Well, I know for a long time I have been describing my particular brand of character immersion as an intense, cathartic connection with the character in which I feel the character's emotional state acutely, understand the mental process of the character acutely and objectively (rather than the character understands it: subjectively) and feel a vicarious emotional response of my own towards the character.
When I look at this in relation to the terms, I know Catharsis to be my primary goal: It is the place that the intense connection to the character is formed, in which I feel, simultaneously, the character's emotional state, my own emotional state, my character's inner workings, my own inner workings and my empathy for the character. Catharsis will make me physically weep when my character's lover dies in her arms even if she does not shed a tear, because while I feel her emotional state as acutely as she does, I am feeling it vicariously. I am immersed in who she is, but I am not her. The feeling of exaltation or relief is something I can validate. An intense, cathartic immersion experience can leave me feeling a little high in an emotionally-induced endorphin way. This goal, IMHO, is all about feeling (For you following the MBTI stuff, it is an immersive F gamer's playground).
I also know, although it is not part of my description above, that Kairosis is a frequent goal for me. It here that I go to for the moments of resounding transition; the moments that feel as if the soundtrack on the drama has picked up and the character's life and the story will never again be the same. The "right moment" of Kairosis is the one where the character and the story interact and change each other, powerfully and irreversibly. This one is both about thinking and about feeling. In order to do this reliably and intentionally it requires a thinking setup, but transitions to feeling mode in the actualization of the moment's resonance. I suppose it is possible to be setup and actualized both in T mode, but I'm not sure if it would lead to the immersive integration that the goal is looking for. This kind of immersion could serve story socket players as or even more effectively than character socket players. It is also, I believe functionally incompatible with Kenosis.
Kenosis is not a goal of mine, but one that is associated with the term immersion quite frequently in discussion, especially in association with larpers over the pond. Also called "Deep IC" or "altered state flow" or that I have been calling "submersion", the goal here is to feel completely like the character and to feel as little like yourself as possible. The feeling of timelessness or transcendence is something that a lot of these folks talk about, even sometimes going so far as to compare it to a religious experience. Again, this is also about Feeling, I think (MBTI note: and I would think that it is commonly a goal of "SF" immersive gamer types, who would require strong myth to make a full transition from self to character). Note: This kind of immersion goal would also work as well for a setting socket player as a character socket player: the goal would be to get out of the player's world and into the world of the player's character.
I also think it's interesting to note that when looked at this way, it's unsurprising that there is so much debate about the compatibility of the goals of nar games and the goals of immersion. A goal like Kairosis requires intentional dramatic framing and intention to achieve the synthesis of character transitioning in the right moment and as such would be perfectly compatible, whereas a goal like Kenosis may repel such deliberate constraint, or force the player back into his own head, making the styles incompatible.
Also, it gives some good groundwork for why immersive players are at odds as to what kinds of game processes or mechanics are counterintuitive to their immersion activity. A Kairotic Immersives might not have trouble discussing stakes or out of game strategy to optimize the “right moment”, Cathartic Immersives might have no trouble authoring to intensify drama but could have real trouble any time the game required transition from a Feeling to Thinking mode, such as crunchy calculation or resource management. Kenotic Immersives might find any out of game negotiation that draws them out of character unappealing.