That’s the “Augustine”. Only other of that name of any consequence dear old Augustine of Canterbury – Apostle to the English –converter of the heathen Anglo-Saxons.

I’d forever counsel caution with regard to merely cut’n’pasting second(third…fourth…)- hand accounts of sources that are themselves translations.
The manuscript history of Tacitus is not entirely clear: some would argue (as in the much more hotly debated passages in Josephus) that they are later interpolations.
There are two first century/ early second century sources: Tacitus, of course, writing perhaps 50 years after events (and with a very strong bias against the Caesars) mentions Pilate and Nero’s persecution. He seems to have regarded Christianity as a purely Jewish sect.
And Suetonius (writing perhaps 70 years after the events) who speaks of the Jews in Rome being aroused by “Chrestus” and consequently expelled by Claudius. He also mentions Nero’s treatment of Christians – which he describes as a “new and evil superstition!
It is not inconceivable that Suetonius is largely repeating Tacitus.
We assuredly cannot simply equate “Chrestus” with “:Christus”. Nor can we be sure that either historian really knew any more than a name or two.

Incidentally, it’s a never-ending source of entertainment to see how rapidly a religious thread can be hi-jacked. Not a mention of the Trinity for a page now.