This is a good question because the difference in the verbs highlights the difference between demons and the devil in the New Testament.
Demonized, having a demon, in an unclean spirit.
Typically where the English versions have “possessed by demons” (or in the King James “possessed by devils”) the Greek word is simply “demonized”, daimonizomenos, “demon-possessed”.
Occasionally the verb used is “have”, such as “he has an unclean spirit” (Mark 3:13), or “demonized, and having the Legion” Mark 5:15, and also Acts 16:16 speaks of a girl who “had” a spirit of Python. Or alternatively without the verb simply “in an unclean spirit” (Greek ἐν πνεύματι ἀκαθάρτῳ – en pneumati akathartoi).
But none of these possession verses ever speaks of anyone being directly possessed by “the devil” himself.
But not having the Devil himself.
So what we see is only one instance of “oppressed by the devil”. And that describes not one individual but a collective of all those demonized, sick, or having unclean spirits whom Jesus healed. It is a collective description of all those Jesus healed, taken together.
The verb katadynasteuō very literally means to oppress. We can see this in James 2:6 where the rich oppress the poor. The only other instance is in Peter’s speech at Caesarea before baptising the first Gentile Christian, Cornelius:
How God anointed Jesus of Nazareth with the Holy Spirit and with power: who went about doing good, and healing all that were oppressed by the devil; for God was with him. (Acts 10:38)
Peter’s explanation of Jesus’s healings of the demon possessed
So what we see here is commentary by Peter on the healings of the demonized by Jesus found in the three synoptic gospels. Peter is drawing together a line from these healings to see this as Jesus liberating the people from the power of the devil or Satan. That theologically connects simple healings and exorcisms with Jesus’s triumph over sin.