Let's have another look at the sentences:
And then she said to me that she was my cousin.
And then told me that she was my cousin.
To me they are absolutely the same.
Said to smb.
Tell smb.
The difference is just 'to say' requires a prepositional indirect object whereas 'to tell' needs an indirect object without a preposition. But this is just a formality. Just a form rather than any change in the meaning of the sentence.
Let's return to the quotation from Cambridge:
"Say focuses on the words someone said and
tell focuses more on the content or message of what someone said"
Frankly speaking this is just an abrakadabra for me. One case focuses on the words and another focuses on the their content.
But you say that you really feel the difference between say and tell when the addressee of the message is mentioned. By the way Cambridge doesn't mention an addressee in their example about 'to say'. Maybe that is the difference. Without the addressee 'to say' means just a fact of sayig. He said, and that is a fact. Whereas 'to tell' means that he conveyed a message to somebody known to the listener or reader. This is a bit another shade. But the only difference is that the addressee can't be omitted in case of 'to tell'.
Here is another article about 'to say' vs 'to tell':
Definition of tell verb in Oxford Advanced Learner's Dictionary. Meaning, pronunciation, picture, example sentences, grammar, usage notes, synonyms and more.
www.oxfordlearnersdictionaries.com
Examples from Oxford:
The doctor told me (that) I had to stay in bed.
OR
The doctor said (that) I had to stay in bed.
NOT
The doctor said me to stay in bed.
As we can see Oxford admits to tell and to say. They seems to be equally reflect the situatuio.
'The doctor said me to stay in bed' is just grammatically incorrect.
'The doctor said to me to stay in bed' is grammatically correct if I'm not mistaken. In this case I can make a conclusion that everything is equal.
But Cambridge says that one case is for words and the other is for contents of the words. One is for contents. The other is for words. Rotate three - five times. Which one is for words?