Luke 4:16-22 is
"Then Jesus came to Nazareth, where He had been brought up. As was His custom, He entered the synagogue on the Sabbath. And when He stood up to read, 17 the scroll of the prophet Isaiah was handed to Him. Unrolling it, He found the place where it was written: 18 “The Spirit of the Lord is on Me, because He has anointed Me to preach good news to the poor. He has sent Me to proclaim liberty to the captives and recovery of sight to the blind, to release the oppressed, 19 to proclaim the year of the Lord’s favor.” 20 Then He rolled up the scroll, returned it to the attendant, and sat down. The eyes of everyone in the synagogue were fixed on Him, 21 and He began by saying, “Today this Scripture is fulfilled in your hearing.” 22 All spoke well of Him and marveled at the gracious words that came from His lips. “Isn’t this the son of Joseph?” they asked."
At this point, Peter has not joined Jesus' ministry. But at Matthew 16:13-16 we have
"When Jesus came to the region of Caesarea Philippi, He questioned His disciples: “Who do people say the Son of Man [i.e., the Jesus] is?” 14 They replied, “Some say John the Baptist; others say Elijah; and still others, Jeremiah or one of the prophets.” “But what about you?” Jesus asked. “Who do you say I am?” 16 Simon Peter answered, “You are the Christ, the Son of the living God.” Jesus replied, “Blessed are you, Simon son of Jonah! For this was not revealed to you by flesh and blood, but by My Father in heaven."
The Pulpit commentary on 16:14 and the list of possibilities for Jesus is interesting (for ex., how could Jesus be John the Baptist, a near contemporary?).
"John the Baptist. This was the opinion of Herod Antipas (Matthew 14:1, 2), who fancied that Christ was animated by the spirit of John the Baptist, or was actually that personage' revived; though it was noticed by others that John did no miracle (John 10:41), and lived a life in contrast to that of Christ (Matthew 11:18, 19). Elias; Elijah, who was taken up to heaven without dying, and was announced by Malachi (Malachi 4:5) as destined to return before the appearance of Messiah. Jeremias. Some opined that he was Jeremiah, who was expected to come as a precursor of Messiah (2 Esdras 2:18), and reveal the tabernacle, ark, and the altar of incense, which, according to the legend of 2 Macc. 2:4-7, he had hidden in Mount Nebo, "until the time that God gather his people again together, and receive them unto mercy." One of the prophets. One of the celebrated prophets of antiquity revived, restored to life again to prepare the way for the great consummation. The well known prediction of Moses (Deuteronomy 18:15) may have given rise to this idea. The four popular opinions here mentioned showed two facts - that Jesus had a high reputation among his contemporaries, and that he was by none at this time regarded as the Messiah. Even those who, after certain of his marvellous works, had been ready to honour him with that title, soon cooled in their ardour, and, checked by his reserve and the slanders of the Pharisees, learned to see in him only a wonder-worker or a precursor of the expected Prince and Liberator."
It seems people did not at the point of the exchange at Matthew 16:13-16 even consider Jesus was the Christ, the Son of God. It is only by direct divine revelation that St. Peter answers that Jesus is the Christ, the Son of God.
So, when Jesus said "Today this Scripture is fulfilled in your hearing." would the people in the synagogue have understood him to be claiming to be the Christ, the Son of God, or would they understood this as another claim, such as those opinions recounted by his apostles (Elijah, and so on)?