Technically a medallion is just a large medal. How large? Good question. "Medal" is way more commonly used. Why Jack is insistent on correcting Annie, I can't answer. More context is probably needed. I'd say Jack is being a bit ridiculous. Haha It's pretty trivial unless you're in a very specific conversation about the difference.

A medallion is another word for medal. It could be used to refer to a small medal or to something worn on the body merely for decoration rather than as a symbol of an achievement. In the story, there must be a particular reason why Jack corrects on this point. The story should explain why. We can't guess reliably unless we read the story.

The reason that Jack tells his sister that it is a medallion, is because it says so in the story. After that, in the Dark woods chapter of the second book, he does mention it three times. A medallion is something that is used for personal usage. It looks like a medal but it larger. The medal is smaller (the size of a coin) and used as a means for achievement (like in sport). Hope this helps, even though it is quite late.

Thank you, all of you! I can understand the basics of the difference. I'll read the story more, expecting that Jack will tell us why he correct his sister every time. :)

Normally medal is used if it wad awarded to mark some achievement. A medallion is not normally awarded. In this case I would suspect that he wanted to be clear thst he did not win it, maybe just bought it as a piece of jewellery.