From my point of view, there's three parts to being a Wizard. There's the mechanical part - which is a matter of training. There's the strength part - which is a matter of birth and/or sponsorship. And there's the sociopolitical part.
Harry had two-thirds of the wizard triumvirate before joining the White Council. He was trained by DuMorne before becoming "official". His training was completed by McCoy, sure, but it certainly wasn't started by him... And he was definitely a qualified apprentice if not a full wizard before then.
Elaine is, by most standards, a better wizard than Harry - it's repeatedly mentioned that she has less raw strength but more refinement and control. Call her better on part "A" (the mechanical bits) and worse on part "B" (raw strength).
She, however, is not a full member of the Council, going so far as throwing the aptitude tests to remain considered a minor practitioner.
... But I'd still consider her a Wizard.
Or, consider
Mavra; Harry practically soiled his pants considering what he described as a Vampire Wizard... But he did apply the term to her, death curse and all.
Step back from the situation for a moment, and consider the White Council as a guild or union. My father is a member of no guilds or unions. By profession, he's a computer scientist working at IBM. He programs, he designs, he studies, he proposes, all technologically-based. And yet when working on any house or structure, I'd trust him more than 95% of the contractors and professionals I've ever met for carpentry and electrical work. He built the front steps (20 years ago), which have survived better than the "professionally" installed steps to the garage. He built the back deck. He built the pool's deck. He installed the french door from the dining room, took a wall out between the dining room and kitchen while re-finishing them, completely remodeled the bathroom including taking out the closet, replacing two doors with one slightly larger one...
You get my point.
You can have all the skills, all the abilities, all the mechanics and strengths of being a Wizard without being a member of "the guild". The White Council polices magicians, enforces rules and regulations, offers protection, inflicts enemies, conducts treaty negotiations, etcetera, etcetera... But they're nothing more than the sociopolitical entity that exists surrounding the idea of wizards. We *know* there are wizards outside the group.
