Hello there! Afri can actually refer to Africans in general, whereas Poeni specifically denotes Phoenicians and Carthaginians. The continent itself was simply known as Africa in Latin and Libya (Λιβύη) in Greek, though Libya could also collectively describe the various peoples living between Mauretania and Tripolitania whom the Carthaginians subjugated. The Phoenicians and later the Carthaginians may have employed similar terminology; in a Neo-Punic inscription from Roman Lepcis dating to 15-17 A.D. (during the reign of Tiberius), the proconsul Lucius Aelius Lamia calls his province "the land of the Libyans" or šd lwbym. The Hebrew analogue of lwbym, לובים, appears in he Biblical Book of Nahum (3:9), dating much earlier to the seventh-century B.C. So it seems likely that Africa was "Libya" for the Phoenicians. :)

I don't know the specific answer to your question, but I wanted to clarify something: in Classical times, Africa was not the name of a continent, it was the name of a very specific province in what is now basically Libya and Tunisia. The concept of continents didn't quite exist in the same way, even if the names Europe, Asia and Africa were already used. I don't believe there was a word that referred to the collective area of Africa as distinct from Asia and Europe.