I am curious about the correct usage of '있다' in an indirect quotation.

Hello! I have a question about the usage of '있다' in indirect quotations. In Korean, '있다' and '계시다' both mean 'to exist' and are verbs. On the other hand, '있다' and '있으시다' mean 'to possess' and are used as adjectives. As far as I understand, when talking about 'existence' in an indirect quotation, it seems correct to use '있는다고 했어요', and when talking about 'possession', it seems correct to use '있다고 했어요'. However, my logic seems incorrect in two examples: 1) '손을 다친 적이 있다고 했어요', 2) '김치를 안 먹는 사람도 있다고 들었어요'. In these cases, both use '있다고'. I would like to know why this difference occurs.