Olorin was only on Middle Earth embodied as a Man, when he first "Died" he would've returned to Valinor, and then been taken out of the realms of the world (If that's what happened).
We know that originally, the Istari were embodied and incarnated. This gave them physical bodies vulnerable to pain, etc.:
For with the consent of Eru they [the Valar] sent members of their own high order, but clad in bodies of as of Men, real and not feigned, but subject to the fears and pains and weariness of earth, able to hunger and thirst and be slain; though because of their noble spirits they did not die, and aged only by the cares and labours of many long years.
Unfinished Tales Part 4 Chapter II: "The Istari"
By 'incarnate' I mean [the Istari] were embodied in physical bodies capable of pain, and weariness, and of afflicting the spirit with physical fear, and of being 'killed', though supported by the angelic spirit they might endure long, and only show slowly the wearing of care and labour.
The Letters of J.R.R. Tolkien 156: To Robert Murray, SJ (Draft). 4 November 1954
We know that the Maiar do not necessarily possess a physical form, however can wear it as a raiment. This would imply that when Gandalf the White returned to Valinor, and his power was restored (the loss of power is discussed later), he would regain his spiritual form and would only take physical form if necessary.
[T]heir shape comes of their knowledge of the visible World, rather than of the World itself; and they need it not, save only as we use raiment, and yet we may be naked and suffer no loss of our being.
The Silmarillion I Ainulindalë
The Istari's memory of the Blessed Realm seems to imply that although their embodied as Men, their Maiar spirit (Fëa) remains within. This causes them to long for Valinor and stick true to their task (which we could see from Saruman's death).
[T]hough they knew whence they came the memory of the Blessed Realm was to them a vision from afar off, for which (so long as they remained true to their mission) they yearned exceedingly. Thus by enduring of free will the pangs of exile and the deceits of Sauron they might redress the evils of that time.
Unfinished Tales Part 4 Chapter II: The Istari
In conclusion, if Gandalf (Olorin) was removed from Time, it was simply for Eru to strengthen him, and when he returned after defeating Sauron would return to being the Maiar Olorin.