Luke omitted several incidents here that the other evangelists included (Matt. 14:22-16:12; Mark 6:45-8:26; John 6:16-66). By doing so, he tied the questions of Herod and the multitude about Jesus' identity with Peter's answer to that question. This selection of material helps the reader see that the question of Jesus' identity was very important to Luke.
9:18-19 The fact that this incident happened near Caesarea Philippi (Matt. 16:13; Mark 8:27) was insignificant to Luke. He may have viewed it as a distracting detail even though the event transpired in Gentile territory.
However, Luke alone mentioned that Jesus was praying. He may have done so to tie this incident to the feeding of the 5,000 when Jesus also prayed (v. 16). Thus he presented the feeding and the revelation to Peter as coming in answer to prayer. Jesus' exemplary dependence on His Father is one of Luke's unique emphases (cf. 3:21; 6:12; 11:1; et al.). He showed Jesus praying before many important events in His ministry. He was evidently praying privately, though the disciples were with Him (cf. 11:1).
Jesus focused attention on the crucial issue of His identity with His question. He wanted the disciples to tell Him who the crowds (Gr. ochloi, the uncommitted masses) believed Him to be. He meant what role did the people believe He fulfilled. The disciples responded with the views that Luke had already revealed (cf. vv. 7-8).
9:20 Speaking for the other disciples Peter answered that Jesus was the Messiah whom God had sent (Ps. 2:2; Dan. 9:26; cf. Isa. 9:6-7; 11:1-16). In saying this Peter rejected the notion that Jesus was just a prophet, even one of the greatest prophets. This is how Moslems view Jesus today. Rather he believed that Jesus was the Messiah promised in the Old Testament.
It is not difficult to know just what Peter's concept of the Messiah was when he made this confession of faith. When Peter's brother first invited him to come and see Jesus, Andrew referred to Jesus as the Messiah (John 1:41). However, most of the Jews of Peter's day believed that the Messiah would be a descendant of David who would overthrow the Romans and establish the kingdom of God on earth. They did not view Him as deity. Matthew recorded Peter's full confession including, "the Son of the living God"(Matt. 16:16). This is a clear statement of Jesus' deity. Why did Luke not include that phrase since it would have clarified what Peter meant? Probably he did not see that as necessary since the title "Christ"had become synonymous with a divine Messiah among Gentiles to whom Luke (and Mark) wrote (cf. 1 John 5:1). Thus Luke appears to have assumed that his readers would understand Peter's confession of Jesus as the Messiah as a confession of His deity.