And yes, the Puritan church was very very much involved.
But that's the thing, there's no one Puritan church, so referring to one capital C Church without specification is like saying "the Hardware Store" when accusing one's local hardware store of selling low-quality goods. There's also no indication they're talking about Salem specifically.
I'm not sure about Europe, but I think those were Catholics.
They were both Catholics and Protestants (it seems areas strongly contested by both sides saw the most trials, though the trials weren't used as a weapon in the conflict between them or anything like that). However, it was decidedly not the Catholic Church, whose position on witchcraft was essentially unchanged from the Middle Ages and didn't allow for witch hunts, which is why areas under strong Church control saw the fewest witch trials. For example the Spanish Inquisition only executed two people for whichcraft throughout the whole period. Like I said local secular courts were the main drivers of the killing; this was a bottom-up affair.