The debate over the charismata is complicated by a lack of standardized terminology. I'm in no position to legislate uniform usage. I'll simply draw some distinctions I think our terminology ought to reflect, and discuss how I think some of the terms ought to be used.
i) In many cases, it would be preferable to use a neutral term like noncessationist as the antonym for cessationist. That doesn't carry any baggage. That doesn't prejudge what noncessationism stands for. It's just a minimal descriptor. A negation of cessationism.
ii) From what I've read, "Pentecostal" usually designates a member of a traditional Pentecostal denomination. In addition, it can reflect an earlier stage of charismatic theology, in the history of the charismatic movement. That includes beliefs like sanctification as a second work of grace, and Spirit-baptism as a third work of grace. Tongues defined as Spirit-baptism. Tongues as the gateway gifts. All Christians ought to speak in tongues, and that, in turn, gives access to the other charismata. Many later charismatics have ditched that paradigm.
iii) One potential distinction between a charismatic and a continuationist is that a continuationist can be a cessationist in practice, but a continuationist in theory. That is to say, a continuationist may believe in the currency of the charismata on purely exegetical grounds, rather than personal experience.
iv) By contrast, I think a charismatic is someone who, by definition, lays claim to having experienced one or more of the charismata.
In addition, charismatics think Christians should actively seek or pursue the charismata. By contrast, a continuationist is merely open to the charismata. Takes a wait-and-see attitude.
Ironically, cessationists interpret 1 Cor 12:31 & 14:1,39 the same way charismatics do. They just think that's moot.
v) Likewise, I think charismatics expect the charismata to be more prevalent. Or at least they think they ought to be prevalent, whereas continuationists are more noncommittal on their frequency.
vi) "Cessationism" has some ambiguities as well. For instance, Calvin was a cessationist respecting Jas 5:14-16. He took the same approach to Jas 5:14-16 that contemporary cessationists take to 1 Cor 12:31 & 14:39.
By Calvin's yardstick, many modern-day cessationists are continuationists respecting Jas 5:14-16. They don't think that's a thing of the past.
On the other hand, contemporary cessationists accommodate Jas 5:14-16 by redefining a miracle so that divine healing in that context is providential rather than miraculous. An answer to prayer, however, extraordinary or supernatural, is merely providential rather than miraculous.
Likewise, classical Dispensationalist Merrill Unger originally took a cessationist position on spiritual warfare in reference to Christians, but later changed to a more continuationist position.
vii) Cessationists typically use 1 Cor 12 as a classification scheme, then fit other Biblical references (e.g. Acts 2) into that preexisting schema. This means other non-Corinthian phenomena like exorcism aren't classified as charismata in cessationist usage. Likewise, this means healers as well as revelatory dreams and visions are automatically reclassified as "gifts."
viii) From my reading, cessationists usually limit the charismata to apostles or those on whom apostles laid hands. They think the charismata were exclusively transmitted through the apostolic imposition of hands. And they confine the function of the charismata to authenticating miracles.