You're right that there is technically some ambiguity, although slightly wrong in your interpretation. In a sentence like ６時に起こしてください it's basically implied that you mean yourself, as in the provided translation ("Please wake me up"). Without context, though, there's no way to be 100% sure. It could be a request to wake up any other person (except the person to whom the request is directed, as in "please wake up at 6").
起こす is a transitive verb, so you're asking this person to wake someone up. Wake me up, wake Jim up, wake Mary up, anyone. In Japanese this would be more explicitly stated with を, as in "６時にジョンを起こしてください。"
If you want that person to get up at 6, you would use ６時に起きてください. 起きる is an intransitive verb, so it has a meaning closer to the simple "wake up" in English, as in "I woke up at 6." Notice the lack of a direct object. "I woke up at 6" does not imply that you woke someone else up -- it only refers to you, the speaker.