The council of elders merely nominates a king (if there are no successors). Irene "identified" Chameleon's talent as being of Magician caliber. Is that sufficient?
EDIT : Could a magic sniffer be used in some way for this purpose?
It's been a long time since I have read any of them, but I remember, from one of the original trilogy, that the ability to resist, defy, or work around a Magician's power identifies you as a Magician yourself. It was in the book where Bink was exiled for not having an identifiable power, because it hid itself. (A Spell for Chameleon)
I think I've got the order - The Elders of Xanth make a judgement on a talent - The King decrees it to be so
This is based on the following passages from "Dragon on a Pedestal"
When Irene had been a child, the Elders of Xanth had judged her magic talent to be excellent, but beneath Magician level
Arnolde Centaur had assumed the throne and decreed Irene's talent to be Magician-level
Talents are usually measured by power (intensity) and usefulness.
To be considered a magician one must have a talent that is powerful, and useful.
Some examples:
There is a character whose talent is to travel back in time a very brief time. This is a very useful ability (time travel) but not particularly powerful (limited range). So she is not considered a magician.
Another example is Dor who is considered a magician because he can speak to any inanimate object vs Grundy who can speak to any living thing. In this context their powers are of the same magnitude, but Dor's is just more useful (he speaks to things that can't already communicate somehow).
The title of Magician is mostly honorific and given to those whose talents have been identified to be both powerful, and useful. There really is no standard or governing body, due to the variety and confusing nature of most talents.
The specific distinction about Bink was that his talent was powerful (no known way of breaking it and no level of magic too strong for it) and useful. The reason why his identification is important is not that it was up for debate as to if it was sufficient for a talent, but rather the question existed as to whether he had a talent at all.
In the end most Magicians are such because the general populace acknowledge and respect the talent as magician quality.