When talking about Jesus and the early Christians, Josephus does not use the New Testament vocabulary we are familiar with
because as we know, he was not a Christian. However, if you search Josephus’ writings for evidence of his opposition to the Christians, you will find it. Josephus attacks Jesus and Christian tradition in many ways.
In his Antiquities of the Jews (Ant.) 18, 1, 1-6, Josephus gives a report about the Jewish sects, where the Christians are not mentioned.
However, the Christians only appear to be missing. For Josephus, the Christians were part of the rebellious movement of Judas the Galilean, and it was therefore unnecessary to mention them specifically. Compare Peter’s denial of Christ in Mark 14:66ff, where Peter is identified in v. 70 as the Galilean (= follower of Jesus = follower of Judas).