In this case, it's the fourth sense of the definition that you quoted:
- something that is published, esp. a periodical.
It's not being used literally. It doesn't mean that she held a newspaper in her hand and spoke to it. Instead, she was speaking to somebody who represented the newspaper.
In a previous paragraph from the same story that you cited:
“I have made the decision to go into a proper therapy [program] in the next few weeks, but it has to be in the UK because I am very, very British and I know that will work best for me,” she told The Sun in a statement on Sunday.
So, the publication in question is The Sun. Rather than repeating The Sun again, the author chose to use a common noun instead—in the same way that pronouns are used in place of names.
Note that saying "she told The Sun" is also not to be taken literally. As with "she told the publication," "she told The Sun" just means that she spoke to somebody working at the company.
The company has the same name as the actual newspaper itself and is being used in a general sense. You could consider it to be an idiomatic use of language.
If you don't give me money to keep quiet, I'm going to go to The New York Times.
In a similar fashion, this means that the person is going to walk into the office building of the company The New York Times which produces the newspaper The New York Times. (And, by implication, talk to somebody there in order to make known some piece of scandal.)
It would actually not be idiomatic to say "The New York Times newspaper agency" or "The New York Times publishing house" because we use a shorter form of expression in normal communication.