Generally, native speakers (Americans anyway) would more often use 'on' than 'in' for sentences like this. But either 'in the morning/evening' or 'on the morning/evening would be ok, but definitely 'on' the same day you move out.
The sentence (When I said "the day after you move out", I meant:) doesn't really make sense the way you've written it. Your question should be rephrased to something like, "If I say, 'the day after you move out,' it would mean:'"
1) In the same day you move out
2) after you move out
In 1), use 'on' not 'in'. On the day you move out, not 'in' the day. I know that's confusing but think of the whole day as a day 'on' the calendar, and in the morning as a period of time in the day (in is about boundaries/ on is about contact - in physical as well as abstract senses)
Neither of the answers you provided would be totally correct.
'The day after you move out' = it's the following day, not the same day.
I see what your thinking here but it doesn't work that way. You'd have to say it differently to get the meaning you want. Something like, 'Later in the day, after you've moved out, I will return your deposit'.
You're playing around with time relations and you need to understand the perfect tenses to do this correctly.
2) 'After you move out' :1 hour, one day, 1 week are all 'after you move out' but your meaning is not open like this. It is either on the same day, or on the following day. After you move is not specific at all.