Seems like it's time to revisit The Hammer of Witches compiled by two Dominican friars and approved by Pope Innocent VIII.

Question #18 may be effective in answering the unbelievers.

MALLEUS MALEFICARUM

[THE HAMMER OF WITCHES]

THE FIRST PART

THE FIRST PART TREATING OF THE THREE NECESSARY CONCOMITANTS OF WITCHCRAFT, WHICH ARE THE DEVIL, A WITCH, AND THE PERMISSION OF ALMIGHTY GOD

Question I..........Whether the Belief that there are such Beings as Witches is so Essential a Part of the Catholic Faith that Obstinacy to maintain the Opposite Opinion manifestly savours of Heresy.

Question II..........If it be in Accordance with the Catholic Faith to maintain that in Order to bring about some Effect of Magic, the Devil must intimately co-operate with the Witch, or whether one without the other, that is to say, the Devil without the Witch, or conversely, could produce such an Effect.

Question III..........Whether Children can be Generated by Incubi and Succubi. Question IV..........By which Devils are the Operations of Incubus and Succubus Practised?

Question V..........What is the Source of the Increase of Works of Witchcraft? Whence comes it that the Practice of Witchcraft hath so notably increased? Question VI..........Concerning Witches who copulate with Devils. Why is it that Women are chiefly addicted to Evil superstitions? Question VII..........Whether Witches can Sway the Minds of Men to Love or Hatred.

Question VIII..........Whether Witches can Hebetate the Powers of Generation or Obstruct the Venereal Act.

Question IX..........Whether Witches may work some Prestidigatory Illusion so that the Male Organ appears to be entirely removed and separate from the Body.

Question X..........Whether Witches can by some Glamour Change Men into Beasts.

Question XI..........That Witches who are Midwives in Various Ways Kill the Child Conceived in the Womb, and Procure an Abortion; or if they do not this Offer New-born Children to Devils.

Question XII..........Whether the Permission of Almighty God is an Accompaniment of Witchcraft.

Question XIII..........Herein is set forth the Question, concerning the Two Divine Permissions which God justly allows, namely, that the Devil, the Author or all Evil, should Sin, and that our First Parents should Fall, from which Origins the Works of Witches are justly suffered to take place.

Solutions of the Arguments.

Question XIV..........The Enormity of Witches is Considered, and it is shown that the Whole Matter should be rightly Set Forth and Declared.

Question XV..........It is Shown that, on Account of the Sins of Witches, the Innocent are often Bewitched, yea, Sometimes even for their Own Sins.

Question XVI..........The Foregoing Truths are Set out in Particular, this by a Comparison of the Works of Witches with Other Baleful Superstitions.

Question XVII..........A Comparison of their Crimes under Fourteen Heads, with the Sins of the Devils of all and every Kind.

Question XVIII..........Here follows the Method of Preaching against and Controverting Five Arguments of Laymen and Lewd Folk, which seem to be Variously Approved, that God does not Allow so Great Power to the Devil and Witches as is involved in the Performance of such Mighty Works of Witchcraft.

No need to go to this much trouble to determine just who is a witch and who isn't. For your edification:

CROWD: A witch! A witch! A witch! A witch! We've found a witch! A witch! A witch! A witch! A witch! We've got a witch! A witch! A witch! Burn her! Burn her!
VILLAGER #1:
We have found a witch. May we burn her?
CROWD:
Burn her! Burn! Burn her! Burn her!
BEDEVERE:
How do you know she is a witch?
VILLAGER #2:
She looks like one.
CROWD:
Right! Yeah! Yeah!
BEDEVERE:
Bring her forward.
WITCH:
I'm not a witch. I'm not a witch.
BEDEVERE:
Uh, but you are dressed as one.
WITCH:
They dressed me up like this.
CROWD:
Augh, we didn't! We didn't...
WITCH:
And this isn't my nose. It's a false one.
BEDEVERE:
Well?
VILLAGER #1:
Well, we did do the nose.
BEDEVERE:
The nose?
VILLAGER #1:
And the hat, but she is a witch!
VILLAGER #2:
Yeah!
CROWD:
We burn her! Right! Yeaaah! Yeaah!
BEDEVERE:
Did you dress her up like this?
VILLAGER #1:
No!
VILLAGER #2 and 3:
No. No.
VILLAGER #1:
Yes.
VILLAGER #2:
Yes.
VILLAGER #1:
Yes. Yeah, a bit.
VILLAGER #3:
A bit.
VILLAGER #1:
She has got a wart.
BEDEVERE:
What makes you think she is a witch?
VILLAGER #3:
Well, she turned me into a newt.
BEDEVERE:
A newt?
VILLAGER #3:
I got better.
VILLAGER #2:
Burn her anyway!

If witchcraft only involves the Devil, the witches and the tacit permission of God (Who allows us to choose Evil so that we have a truely Free Will), then the magic talked about in the Harry Potter books cannot be witchcraft, because the Devil is no-where present. Indeed, worship is no-where present in those books, that I know (and I certainly have not wasted my time reading the entire series of what are, after all, childrens' books), which rules out both Satanic Witchcraft and neo-pagan Wicca. What is taught at Hoggwarts seems to be sorcery, magic as the science of manipulating non-material forces by both lawful algorythms (spells and potions) and even more by force of psychic talents.