A charismatic church is one which is very lively, people raise their hands when singing, and shout out Amen etc. They tend to sing lots of songs, and don't have a formal order of service with set words to follow.
It is similar to a pentecostal church which is one type of church.
There are charismatic churches which are Anglican, Catholic, Baptist etc etc. They are not a cult. Some, can be a bit over the top. The opposite would probably be described as a traditional church, where there is a more traditional, stand up, sing a hymn, sit down type of service.
There is no ''official'' ruling on a cult, but cults do have similar patterns, they tend to be overly focussed on one person, then tend to try and tell their people what they can and can't do (in a controlling way) - eg what you do with your money, who you marry, they tend to be very controlling, and restrict people as to what they can read, don't encourage questions etc. There are some churches (charismatic and not) who tend in this direction, I think they aren't healthy and would avoid them.
I go to a charismatic church. Yes people have strong opinions, and believe certain things, but everyone is encouraged to question, to find things for themselves, to read whatever they want, to basically be a normal person, and make choices for themselves. There are plenty there who don't agree with all that is preached, so they will go and challenge the preacher, and that is normal.
HTH