Unless the state has a definition of an official date, there isn't one.
Instead, it is a question of domicile. Your domicile is where
1. You own a house or have a signed apartment lease
2. Have a landline (not so common any more).
3. Pay utilities
4. Have a drivers license
5. Are registered to vote
6. Get your mail
7. Have told your friends and your employer where you live
8. Is the place that you intend to return to when you have been traveling or working out of town.
You can have only one domicile at a time, and your domicile does not change until you move somewhere else and take positive steps to do the 8+ things above.
As you can see, you can satisfy some but not others of the 8 things about (which in itself is not an exhaustive list).
Basically when asked in an audit, you need to be able to show that you satisfy the majority of these items. Most importantly, you need to convince the auditor that you intended to live in this place.
So what's the date? It is when you say, "I live here", having done a lot of the 8 things.
Having said all this, some states (Virginia come to mind) require you to file as a resident if you have lived or worked in the state for X amount of time (from memory, 183 days in VA). This means that you might be required to file as a resident in two different states, even though you can have only one domicile.
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