Translation:I told her that my cousin deleted that word.
I think that you will have to use present simple tense in English. Present Perfect Tense is used for actions started in the past and still ongoing now. Therefore you cannot use present perfect here. I am not a native English speaker, so anybody may correct me if I am wrong.
It is about the emphasis, I believe. Which word are you stressing? Does it mean "It was my cousin who deleted that word," in which case yours is right, or does it mean, "It was that word that your cousin deleted," in which case Duo's is. The word emphasised goes before the verb. Just my opinion, though.