The in service date, sometimes referred to as the available date, is the earliest date a renter "could" have moved in. This date is at least one day "after" you moved out. Realistically though for most, it's usually about a week after you move out at the soonest. That's because it usually takes a few days to clean things up and prepare the property for showing to perspective tenants. When done, it's move in ready. That's commonly defined as the date you put the "FOR RENT" sign in the front yard.
But in your case, there's no possible way the unit was physically available and move in ready on or before the date you physically moved out regardless of when you put the FOR RENT sign in the front yard. So keep it simple and make your available date, the date the renter actually moved in.
Take note that with your option #2, expenses incurred while you were still living in it are not a deductible rental expense. If you're doing this in order to deduct expenses that would otherwise be start-up expenses (which are not allowed for residential rental property) you risk getting bit down the road starting with an "audit by mail" from the IRS. Now in reality the chances of that are low - but they're also not zero.
