I just wanted to know the difference between the two denominations and ended up confused and my question unanswered
Well my brother Blain, it is a good question.
Pentecostal in this context is not a denomination, neither is Charismatic or Third Wave. In this context we are talking about a theological view or interpretation of scriptures concerning the Holy Spirit, the definition of the Baptism of the Holy Spirit, the operation of the gifts of the Holy Spirit in the church today.
One might even call them movements, in the definitions that were given by the source you posted. At least the definition that was given about the Charismatics and Third Wave. Your source sounded like a reformist trying to explain what pentecostals and charismatics believe, and it is always best to get the definition from the pentecostal and charismatic instead.
There are denominations that use the word Pentecostal in their official name and there are denominations that are pentecostal in their theology but do not use the name pentecostal in their name. So for example the United Pentecostal Church of God In Christ is a denomination that teaches things that most other pentecostal denominations like the Assemblies of God would disagree with. UPC is known for telling people they have to speak in tongues to be saved and most pentecostals do not believe that. The UPC is considered a cult to other pentecostals and because of denominations that use the word Pentecostal in their names but have given pentecostals such a bad reputation (i.e. snake handlers, Oneness doctrine, etc) many today do not want to be called pentecostal so as not to be confused with those extreme groups that use Pentecostal in their denominational church name.
Traditional pentecostalism from a theological view was simply declaring that you believe that what happened on the day of Pentecost started a pattern that is repeated in the book of Acts namely a subsequent experience to regeneration and for the purpose of being empowered by the Spirit to fulfill the great commission. That the gift of speaking in tongues is the scriptural pattern that is repeated and especially with statements like the one in
Acts 10: 45 And they of the circumcision which believed were astonished, as many as came with Peter, because that on the Gentiles also was poured out the gift of the Holy Ghost. For they heard them speak with tongues...
Charismatics in practice are really just people who believe the same thing, that the gifts of the Holy Spirit (charisma) are for today and they were given to the church for the edifying of the body of Christ and did not cease at the writing of Revelation.
Charismatic movement seemed to be what your source was following which was that there was a movement that swept both Catholic and evangelic denominations during the 70s where many received the baptism of the Holy Spirit and speaking in tongues and tried to bring it into their denominations, some of these local churches continued to allow these gifts to be expressed while remaining in their non pentecostal denominations. And yet most of these people also testified to speaking in tongues so it is really a mute point about their not agreeing that speaking in tongues was the initial evidence since they were speaking in tongues while arguing about it.
Today there is really little if any difference between a Pentecostal theology and a Charismatic Theology. I doubt you can find anyone who can point out any difference between a nondenominational church that calls itself Charismatic and a typical Assembly of God chruch that uses both words, pentecostal or charismatic interchangably to say the same thing.
That is the theology that the gifts of the Holy Spirit including, and usually starting with the gift of tongues, is available for every believer today who asks for it.
I believe the word Charismatic has taken over the use of the word Pentecostal for most people because they do not want to have to explain the difference between the theology pentecostal and those other embarrasing denominations (i.e snake handlers) who use the name Pentecostal. So they are abandoning the word for something less confusing. Charismatic works just as well.
Now for the Third Wave, this is a movement that included the Vineyard Churches (John Wimber) and the teachings of Fuller Theologian Peter Wagner, I guess you could call it a revival or extension of the Charismatic movement. It is not really anything anyone claims to be. No one really says. "I'm a Third Waver" it is more of a description of a movement. Not everything that has swept the charismatic churches can be included as belonging to a THIRD WAVE, that is arbitrary labeling.
Maybe the new label is Continuationists? Not that, that is a new term but it is getting new traction as people are wanting to express that they are not a cessationists but they are not one of those snake handlers either.
I still use the word pentecostal because I like to start with Acts 2 and explain why I use that word, and go through all the scriptures on the subject when I teach it and it is the scriptures that give faith for people to believe and receive.
I am pentecostal in my theology, I am also charismatic in my theology. I am also a Continuationists in my theology. All are saying the same thing.
I am not a Third Waver, because that just sounds weird
and no one knows exactly what it means. LOL I certainly do not agree with Kingdom Now Dominion theology, Christian demon deliverance (pigs in the parlor book) or Strategic Spiritual Warfare and naming of chief city demons and prayer walks that bind demons and all the complicated things that go along with what Peter Wagner was teaching in this regard, though I don't discount everything Peter Wagner taught.