From the standpoint of literary criticism, Mark's report of the centurion's declaration, "Truly this man was the Son of God!" -- proclaimed at the end of Jesus' earthly life -- is a coda or reiterations of Mark's opening statement:
Mark 1:1
The beginning of the gospel of Jesus Christ, the Son of God.
Beyond that, in Mark's account we are left to speculate. One idea is that the others who were crucified had not died yet (Mark 15:44). Another is that the centurion did not associate the accompanying earthquake with their death. However, if we include the reports of the other synoptic gospels, the earthquake hypothesis is given added weight and other relevant details also emerge.
Matthew 27:54
Jesus cried out again in a loud voice, and gave up his spirit. 51 And behold, the veil of the sanctuary was torn in two from top to bottom. The earth quaked, rocks were split, 52 tombs were opened, and the bodies of many saints who had fallen asleep were raised. 53 And coming forth from their tombs after his resurrection, they entered the holy city and appeared to many. The centurion and the men with him who were keeping watch over Jesus
feared greatly when they saw the earthquake and all that was
happening, and they said, “Truly, this was the Son of God!”
So in Matthew's account, the centurion's response is directly connected to the earthquake and other powerful events. We also learn from Matthew that Jesus possibly recited a psalm while he died (v. 34). This may have impressed the centurion.
Luke also associates the centurion's response to the circumstances surrounding Jesus' death:
The centurion who witnessed what had happened glorified God and said,
“This man was innocent beyond doubt.”
Luke also adds other details that may have impressed the centurion, such as Jesus saying "Father forgive them for they know not what they do" and the conversation between Jesus and the two criminals, in which Jesus displays confident knowledge about the afterlife. This in particular may have differentiated Jesus from the others in the mind of the centurion.
To answer the OP's secondary question, the centurion's statement (including the verb "was") does convey the centurion's conviction about the identity of Jesus Christ, but it does not necessarily consider "the entire process from betrayal to crucifixion." Rather, it seems to consider primarily the immediate circumstances and manner of Jesus' death.
Conclusion: In Mark, the reason for the centurion's declaration is unclear beyond the fact that the other criminals were probably still alive when Jesus died. But Matthew associates the centurion's statement directly with the earthquake and various supernatural events. Both Matthew and Luke provide additional details that further differentiate Jesus from the other criminals; and these details may also have influenced the centurion's declaration. However, the most certain answer -- supported by all three synoptic reports -- is that the centurion was affected by the earthquake.