Learn Russian: Это мой личный дневник, я пишу исключительно для себя. - This is my personal diary; I write exclusively for myself.

Я пишу один means "I write alone" in the physical sense, meaning nobody else is in the room with you. Я пишу для себя means "I write for myself", meaning you are the only intended audience. In the context of a personal diary, для себя makes the most sense.

Why is it для себя and not для меня since the subject of the sentence is "I"?

In Russian, whenever the target of the action is the same as the subject of the sentence, you must use the reflexive pronoun себя (myself, yourself, himself, etc.). Because "I" am doing the writing for "me", using для меня would sound like a mistake to a native speaker here.

Could I use частный instead of личный if I want to say it is a "private" diary?

Not really. Личный means "personal" (belonging to a specific individual, like personal thoughts or belongings). Частный means "private" in the economic or legal sense, like private property or a privately owned business. For a diary, it is always a личный дневник.

Is the adverb исключительно related to the Russian word for "exception"?