In general:
If you are considered a non-resident of state X, you generally still owe income tax on any income that you earn while physically working or living in state X. You would report that on a non-resident return. If you also file a tax return in the state where you are a permanent resident, you report all your world-wide income no matter where it was earned, and you can claim a credit for the taxes you paid in the other state.
The regulations quoted for Virginia cover two different situations.
1. If you are a domiciliary resident of Virginia for 183 days, you are a Virginia resident for tax purposes even if you have a permanent residence someplace else. The rules for domiciliary resident vary from state to state. Sometimes, an apartment counts but a hotel does not, other times, a hotel also counts. But, as a member of the military, you are not subject to this rule even if you kept an apartment in VA for more than 183 days.
2. Separately, if you earn income from physically working in VA, you owe VA income tax on that part of your income even if you are considered a non-resident.
So yes, you owe a VA tax return for your Uber.
