The part mentioning Jesus’s crucifixion in Josephus is extremely likely to have been altered if not entirely fabricated.

The idea that the historical figure was known as either ‘Jesus’ or ‘Christ’ is almost 0% given the former is a Greek version of the Aramaic name and the same for the second being the Greek version of Messiah, but that one is even less likely given in the earliest cannonical gospel he only identified that way in secret and there’s no mention of it in the earliest apocrypha.

In many ways, it’s the various differences between the account of a historical Jesus and the various other Messianic figures in Judea that I think lends the most credence to the historicity of an underlying historical Jesus.

One tends to make things up in ways that fit with what one knows, not make up specific inconvenient things out of context with what would have been expected.