The big question is the difference between a prologue and chapter 1. Let's say your book takes place over a few weeks, and a chapter is roughly a day. Roughly -- some days span several chapters, some chapters span several days, but roughly. You want us to know something that happened 5 years ago. Or your book spans years, with each chapter a month or so, and you want to show something from 20 years before, or 200. So either you put that in a prologue, or (if you want to reveal it later) you add a flashback or some extensive dialogue in which we come to learn about it. Prologues like this tend to look like this:
- a small child interacts with an adult, perhaps a parent, perhaps a teacher. The child shares their dreams with the adult and is encouraged or belittled. Or the adult tries to teach the child something and the child's internal monologue reveals their dreams, often quite different from what the adult wants. This teaches us the struggle our protagonist is undergoing when the book starts.
- a person lives a nice kind of life - happy in a peaceful village, a happy family, a fulfilling job. (Little do they know it's all going to disappear. The book picks up much later with all of that gone.)
- a government official or a soldier some other authority has a conversation with someone about a problem and how it's going to be dealt with. (This sets up the situation the book people all have to deal with, like there is a war or they have been conquered or there are laws there didn't use to be.)
- someone makes a prediction, or hides a magic weapon, or curses all the descendants of one person, or founds a religion. This lets you provide slightly mysterious backstory and history.
The gap of time between the prologue and chapter 1 should be significant. Years, decades, even centuries. If the book starts Tuesday, a prologue set on Monday is not a prologue.
So, how did the rivalry between D & I start? A prologue set in their childhood might cover that incident. As part of it, the friendship between D and C would also be covered, along with action and dialog that shows us D's personality. How did each of them come to live where they do? Were they all born there, or did one arrive later? Why? How did that go? You could even close the prologue with a prediction like someone saying of D "he's the kind of kid who could grow up to be a murderer!" and another saying "or Prime Minister!" and the two of them laughing. Now we have context for what the adult characters do and say and how they interact.
The paragraph in your question isn't a prologue. Possibly, possibly it could belong in a letter from one character to another -- but it often is hard to believe that teens or your adults or peasant villagers or whatnot would write crisply and well in letters and notes and diaries. Those tend to be ramblier and not to summarize things well. So why not write a prologue set years before the story begins that tells you who the main players are, how they feel about each other (with a glimpse of why) and a bit about what they are like? Have D be cruel, but then witty and fun loving. Set it up. The cool thing about a prologue is you don't have to wrap it up. Just leave off, and then pick up again when the book really starts.