Harry's a wizard because the books define 'wizard' as the thing Harry is.
It sounds like you're using some definitions for "wizard" and "warlock" from outside the Harry Potter franchise. This is a non-starter for two reasons. First, historic terms for magic users have extremely vague definitions and it's only in recent fiction that the terms have come to really settle down into firm notions of different kinds of magic-user. Second, definitions outside the Harry Potter franchise don't matter because the franchise defines its own terms and external definitions need not apply.
Mana doesn't exist in HPverse, so using it as a definition is pointless. Besides, "mana" and "warlock" are words from opposite sides of the world in totally different traditions. "Warlock" is a European word with connotations of devil-worship and diabolical power (NOT specifically life-draining magic), while "mana" is an Austronesian word describing both mystic and social power of a positive or neutral nature.
These words weren't put together until the 1960s or 1970s, when popular fiction appropriated the words while changing the meaning. It's at roughly this point that notions like "magic users only use one kind of magic unless they're really special" came into prevalence--and with those notions, the need for more codified terminology to distinguish between them. However, those distinctions only arose in the kinds of fiction where they were needed; in the wider world of magical fiction such terms tend to retain their older meanings.
Thus, the proscriptive definitions you're working with are specific to a particular narrow subset of modern magical fiction. Applying those definitions outside their intended context is bound to be confusing at best, because the definitions were invented BY and FOR that context.
Because JK has no interest in providing clear explanations of what magic is or where a wizard gets his power, trying to categorise HP casters by the source of their power is a fundamental mismatch--that universe doesn't care about the categories you're trying to impose on it. JK Rowling draws on older mystic lore than video games and the novels of Niven, so her books don't use the concepts and terms of those modern works.
Since all magic-using humans are wizards in her works, 'warlock' is a subset of 'wizard' rather than a uniquely different kind of magician.