You're correct when you say that 또 emphasizes "again" and 다시 "over again from the start".
But the difference is not huge, and I don't think either one has a more positive or negative nuance.
Any positive or negative nuance there might be in 아 귀여워! 또 해요! and 아니! 다시 해요! are mostly from other things and not 또 or 다시 (such as the context, voice tone, and the exclamation "아니!").
또 simply implies that the same/similar thing is happening again - the stress is on the "additional occurrence".
다시 is like something is happening "over again", following a similar pattern as before.
(To simplify it, you might think of 또 as a simple "again", and 다시 as "over" or "(all) over again".)
- 또 만나요. See you again. (light and casual nuance)
- 다시 만나요. (similar, but sounds slightly more serious)
(If you're parting for a long time, or in a grave situation, you'd say 다시 rather than 또.)
- 세 번이나 떨어진 뒤에 또 떨어졌어! After three failures, I failed again! (emphasizes the additional occurrence)
- 다시 한번 말씀드립니다. I am repeating it for you once more.
- 그림을 망쳤어. 다시 그려야 돼. I ruined the painting. I have to redo it over. (또 is not as natural)
- 자 이걸로 송년 파티를 끝냅니다. 내년에 또/다시 만나요! This is it for the year-end party. See you next year. (Both sound perfectly natural)