The term "Christian" was used in New Testament times of the Jews and God-fearers who followed Jesus as their Messiah. The problem arises when we try to transfer the modern meaning of Christian back to the first century.
When we say "Christian" today we usually have an image in mind of a religion that is completely separate from Judaism, and usually antithetical to it. We think of church buildings full of people gathering on Sunday morning with their New Testaments. When we try to transfer this image to the first century, it becomes an anachronism.
Imagine a movie set in Civil War times. If someone in the film mentioned a camera, you would have a distinct image of the kind of camera it would be. But then if he pulled out a cellular phone with a tiny digital camera attached, it would be a shock and have no credibility. That is what an anachronism is.
We are faced with the same situation when we talk about Paul, or any of the apostles, being Christians. To most of us that means they were not Jews, they were Christians instead.
In the first century the word "Christian" implied a Jewish person. That's the only kind of believers there were at the beginning. The word was probably used in a derogatory manner, as a way of making fun of someone. Paul never used it, of himself or anyone else. In fact, the word is only used three times in the New Testament. Once in Acts 11:26 where the disciples were called Christians at Antioch, once in Acts 26:28 when Agrippa asks if Paul is trying to persuade him to be a Christian (Paul, incidentally, avoids using the word in his reply), and once in I Peter 4:16 in speaking about suffering as a Christian.
If we talk about Paul becoming a Christian or teaching about Christian living, we are claiming more than Paul did. And the person listening is likely to get an impression that is far from the truth. Paul lived and taught a Jewish Torah-observant lifestyle following a Jewish Torah-observant Messiah. He and his followers worshiped in synagogues on the Sabbath, along with the rest of the Jewish community. They had no New Testament; their Bible was the Hebrew Scriptures, the Law and the Prophets. Christianity did not exist as a separate religion until very late in the first century, and we need to remember that when using the word regarding early believers.