I'm sure you probably know all this, since we've discussed this before:
Tacitus' work usually came from Roman archives, but occassionally he used other secondary sources. The fact that Tacitus called Pilate "the procurator" instead of "the prefect", as the Pilate Inscription describes him, and that he referred to Jesus as "Christ" suggests that his sources were Christian, not Roman. No Roman archives mentioning Christ have been found.
Suetonius wrote of "Chrestus", which wasn't a reference to Christ. Pliny the Younger simply described the beliefs of some arrested Christians. He doesn't mention anything that suggests Jesus Christ existed.
Josephus' writings are heavily disputed. His description of Jesus (from the passage you haven't quoted) conflicts with his own beliefs. It's suggested that the passage may have been corrupted by Christians. In the passage you have quoted, the phrase "who was called Christ" is disputed, as is the idea that the "Jesus" mentioned is the Biblical Jesus.
Cornelius Tacitus, a respected first-century Roman historian, wrote: “The name Christian is derived from Christ, whom the procurator Pontius Pilate had executed in the reign of Tiberius.”
Suetonius and Pliny the Younger, other Roman writers of the time, also referred to Christ.
Flavius Josephus, a first-century Jewish historian, wrote of James, whom he identified as “the brother of Jesus, who was called Christ.”