Regarding predestination: Many Christians are awed by the word predestination that we forgot about context.
In biblical context, predestination means that God has pre-planned to offer redemption to the Gentiles; it was NOT about predestination of individuals.
For 2000 years before the apostles' times, Yahweh was God to the Jews only, while other pagan nations worship many pluralistic gods. Even after Jesus' resurrection, the apostles initially thought that redemption was for Jews only. However, Peter had a vision about eating unclean food, and they realized that the Gentiles are invited too. Now, this was shocking to the Jews because it went against their tradition that was so rooted in the God of Abraham and Jacob, where Gentiles had no part in. Upset about the drastic change, the Jews demanded that Gentiles followed Jewish customs (many Christian Jews were still practicing circumcision and Sabbath at that time). Amid the hostility, even Peter distanced himself from the Gentiles, and Paul opposed him for that. To assure the Gentiles, Paul explained in Ephesians (and Letter of Romans) that God had always predestined (pre-planned) to offer redemption to the Gentiles. Let me explain the following verses while quoting them:
Ephesians 1:12, 13
[12]"In him we were also chosen, having been predestined according to the plan of him who works out everything in conformity with the purpose of his will, in order that we, who were the first to put our hope in Christ, might be for the praise of his glory.
==> "we, who were the first to hope in Christ" refers to the Jews who had believed in God for 2000 years, since the times of Abraham, Jacob and Moses.
Speaking as a Jew, Paul said the pronoun "we" [v 12], and as he referred to the Gentiles, he said "you" [v 13] ==>
[13]
And you also were included in Christ when you heard the message of truth, the gospel of your salvation
==> The Gentiles, who were previously outsiders, are now included in God's redemption.
When seen in context, the Bible was not talking about predestination of individuals. Instead, predestination means God has always planned -- or pre-planned, predestined or destined -- to offer redemption to the Gentiles. Different translations use different words but when we know context, we will not drill into the words technically.
Regarding the words "chosen people" or "Chosen by God": We tend to interpret factually today, thinking that God handpick or choose who should be saved. However, during biblical times, people were more submissive to God -- much, much more than people today. Back then, they don't say that they chose to believe in Him; instead they said that God chose them. It is a humble way of speaking. To say that they chose God would have sounded arrogant or inappropriate in those times. Today, 2000 years later, times change and languages change too, and we don't speak that way anymore. To us now, "chosen by God" sounds like He literally chooses Jason or Susan. But back then, it was really a humble way to say that it was a privilege to be part of God's people or kingdom.