This is going to be a bit of a different typing thread. Please ignore my Enneagram self-typing, I assure you it is merely incidental.
Allow me to properly preface this thread by saying that I don't think a static approach to typing is compatible with one of the underlying topics of study of the series: whether time changes people or not. I'm going to call in @woofwoofl for his dynamic approach to typing, and anyone else who would care to contribute.
As a quick spoiler:
To begin with, let's look at the characters. One of the structuring arcs to the story is the gradual inversion of the relationship between two of the characters: Madoka and Homura. Homura begins as a socially insecure, shrinking, passive individual who is inspired and uplifted by Madoka's assertive and loving persona. By the time the series occurs, Homura has become an emotionally guarded, highly competent agent, whose constant interference with Madoka leaves her as a shy, passive, but still big-hearted individual.
This manifests the notion that time draws out coherent patterns in people's personalities, even against apparent gross changes. Madoka retains her heart, Homura retains her love for Madoka; both of these characters have dramatic changes in personality.
Meanwhile, another character, Sayaka represents the things that do not change over time. She is always a chivalrous idealist. She is always betrayed by reality. She always falls to despairs due to this betrayal. Mami, on the other hand, is nondeterministic in her behaviour through scenarios. In one case she is betrayed by relaxing into humanity and intimacy; in another she is driven to "merciful" murder, driven by horror and self-hatred.
Another runnng topic of inquiry is love. While Homura's feelings for Madoka are the frustrated romantic's obsession and anguish at separation, the series looks at various different processes. Sayaka is part agent of morality, part doting and demure self-cast servant. I believe she feels awe and inferiority in the face of the boy she adores, and does what she believes is right to care for him. On the other hand, when she despairs, it is because she hates herself for fighting with a dear friend, and questions why she ought to be a brave knight fighting to defend people she finds deeply disgusting. I believe these two episodes work to paint her as a highly moral, idealistic individual with little patience with herself as a less than ideal person.
Kyoko captures the narcissistic aspects of love. In Sayaka, I believe she sees herself, enough so that she fights and fights to reach through Sayaka's despair, failing quite tragically. In Kyoko, (at least according to the third movie), Sayaka must see a similarly betrayed and wounded individual, enough so that she turns from her despair briefly to acknowledge Kyoko's efforts. These two I think are capable of some weird chimeric respecting, pitying empathic connection.
So, what types are they? By the time of the series, and through the third movie:
Homura is a very universal character. She is the frustrated romantic's anguished obsession. Blissful tranquility, elation, rage, self-admonishment, sadness, and selfless dedication. She is a relatable fellow traveller on unpredictable waters. All of her struggles culminate in something that might be, but isn't, desire or insatiability; a thing she sagely captures as the pinnacle of human emotionality, a thing deeper than despair and more brilliant than hope. She is also a nietzsche-flavoured byronic hero, to the extent that she self-destructs by alienating Madoka and casting herself as an "evil", Satanic figure. I want her to be an EIE, but I'm severely biased.
Sayaka, incidentally, is the one that describes Homura's love as being somewhere in the neighbourhood of desire and insatiability. While this does capture Homura's actions, I think it also betrays her own high standards and a very self-sacrificing, ought-to driven personality. My initial suspicion is EII.
As for Mami, Kyoko, and Madoka, I really don't have good handles on their characters, having not paid much attention to them. If you're just here for typings, have some: dunno, SEE, SEI respectively.