Who are the 'People of the Book' described in Q4,171? Are they Jews? Could they be Christian?

The questions were asked by viewers of the videos I (Odon) made with Jay on his Youtube Channel, Pfanderfilms.

Here is my reading of Q4,170-171:
4,170: People of mankind! The apostle [Jesus, as it will be explained thereafter] has brought you the truth from your Lord, so believe! (It will be) better for you. But if you disbelieve [KFR root, which relates to the "bad Jews" who cover the sacred scripture] – surely to God (belongs) whatever is in the heavens and the earth. God is knowing, wise.

4,171: People of the Book [so here again, the preacher talks about Jews, making a connection with the previous verse] Do not go beyond the limits in your religion, and do not say about God but the truth. Verily the Messiah, Jesus, son of Mary, was an apostle of God, and His word, which He cast into Mary, and a spirit from Him. So believe in God and His apostles.
And do not say, ‘Three.’ Cease. Good for you (this way).

I think the preacher wanted to explain that recognizing Jesus as the Messiah should be natural to the People of the Book (otherwise it will be a deviation of the religion, an act of "covering"). And to explain that recognizing Jesus as God's word, born from a virgin and God's spirit is still not beyond the limits of “the religion of the people of the book” as it does not imply to recognize Jesus as God himself. Here, I understand that the preacher was saying something like "Jews, accept Jesus as the messiah, but don’t go so far as to make him God. So no trinity!"

https://thegreatsecretofislam.com/
Source: https://www.youtube.com/watch?v=5Iriu...