The Immortals fall roughly into four categories. Outermost or least "central" would be the "human Immortals", otherwise indistinguishable from humans except for their relatively long, healthy and virtuous lives. More central are the "earth Immortals", who live for an unusually long period of time in the mortal realm, far beyond the normal human lifespan. More central yet are the "spirit Immortals", who are (or have become) genuinely deathless and ever-living in other realms or dimensions neighboring to ours.
Most central are the "celestial Immortals", who most closely correspond to the Powers, or in some cases, to the One. Many of these are considered to be direct manifestations of the universal cosmic energy, or that energy itself become (or made) conscious, so that the beings in question seem to wield the power of gods. Others were thought to have been promoted to this status because of tremendous service done to the One in the mortal and immortal realms.
There were immortals who were the spirits of stars or animals. Some were human beings who had done good deeds and were rewarded with immortality, or achieved it by cultivating body and mind. They were as diverse as any group of human beings might be: among them were not only healers and teachers and social activists, but career politicians, scientists, poets and masters of feng shui, military commanders, entertainers, householders and entrepreneurs. What they all had in common, however, is that they became interested in the innermost workings and welfare of the Universe at an early age; they shunned fame and fortune; and they lived simple and unencumbered lives, whether in public or private practice.
There were numerous Chinese wizards who were mistaken for Immortals...and, in the normal way of things, probably a lot of Immortals who were mistaken for wizards.