Himawari’s (blue girl) part is correct. Chinatsu’s (pink) part I’m not sure about. I read it as “I started reading that!” as in Chinatsu also started reading that book and she is expressing that to Himawari. But your interpretation might be correct, it makes sense given what Himawari says. Deepl takes “それ読み始めたんだ！” as “I started reading that!” but that is with no context. I don’t quite understand how the “ん” in “それ読み始めたんだ！” changes the meaning. I don’t think I understand any more than you unfortunately, sorry.