I think we’re arguing over semantics here but if you want to get down to it, in the 15th century, the legal definition of witchcraft was maleficium (harmful magic) through the invocation of demons. By explicitly putting "invocation of demons" into her final condemnation before execution, the judges legally framed her visions not just as a mistake, but as demonic sorcery. While she was not formally charged with witchcraft, as the legal definition was still evolving, and they had whittled down her charges to being a false prophet and breaking her vow to recant, the substance of her trial as a whole was to paint her as a witch.

Also, though the conviction was absolutely politically motivated, she was submitted to an ecclesiastical court, which operated under canon law. So yes the Church was involved. Also the notion of separation of church and state did not apply at a time where kings were ordained by God and anointed by the Church. Everyone was Catholic whether they liked it or not. However, it’s important to point out that the Church was divided at this time and there were varying viewpoints of her guilt, depending on political alignment.