The Arian Controversy and the Council of Nicea

[TITVS SABATINVS AQVILIVS]

But Constantine defeated Maxentius in AD 311. So, Constantine could not have copied Lincinius, right ? I just checked the time line of events in my books.

The time line is not sure: there are three accounts about Constantine's dreams or visions:

-from the Panegyrici Latini 313; the account is dated 310 or 311 and refers to a campaign against the Franks near Trier (Treviri), or Autun.

-from Lactantius, De mortibus persecutorum XLIV

-from Eusebius, Vita Constantini, I.XXVI-XXVII

The earliest of the three, it's outright decidedly pagan in tone (Apollo, the sol invictus) and, like in the other two, after the vision there comes victory in battle:

“For, O Constantine, you saw, I believe, your protector Apollo, in company with Victory, offering you laurel crowns... “

According to Schaff, the account from Lactantius has been questioned by Burckhardt to have actually been composed by Lactantius himself. Surely, Schaff says, it was composed soon after the event, when Constantine was still on good terms with Licinius, because a similar vision is attributed to Licinius in chapter 46 (Constantine and Licinius agreed on religious toleration in 313):

Tunc proxima nocte Licinio quiescenti adsistit angelus dei monens, ut ocius surgeret atque oraret deum summum cum omni exercitu suo; illius fore victoriam, si fecisset.

Some important differences between the two "dreams": Licinius is guided explictly by an angel, and the angel tells him to pray to the deum summum. On the other hand, Constantine is told to emboss a generic (in the dream) signum dei on the shields. Actually the monogram appears on Constantine's coins only from 315 onwards, i.e. 3 years after the battle at the Milvian Bridge. It is not so clear how "transversa X littera, summo capite circumflexo" can be seen as the Christ's monogram: this sign is then explained to be the symbolum of Christ, but was a symbol used by the Costantinus' gallic soldiers too as solar symbol (as often we see).
The same vision of the labarum cross before the Milvian Bridge battle was reported by Eusebius only in 337 in his “Vita Costantiniâ€