Interestingly, the Wesley's of Methodism are the true pioneers of this movement.
The Assemblies of God considers themselves a Pentecostal movement. Charismatic usually refers to a movement within a non-Pentecostal founded church (e.g. Charismatic Catholics, Charismatic Lutherans) or a movement or denomination that came out of a non-Pentecostal based church.
https://en.wikipedia.org/wiki/Charismatic_Movement