In either of the cases that you mentioned you do not have to report your state or local income tax refund from Form 1099-G box 2. That's assuming that the "previous year" is the year that the refund is for, as shown in box 3 of the 1099-G. It's the year in box 3 that matters. In most cases that's that previous year, but it could be an earlier year.
You only have to report the state or local income tax refund if you claimed an itemized deduction for state and local income tax on your federal tax return for the year that the refund is for. You do not have to report it if you used the standard deduction, or if you claimed itemized deductions but claimed state and local sales tax instead of income tax.
No, you do not need to report a state refund (box 2 of 1099-G) if you used the Standard Deduction last year.
Please see Do I need to report the state or local refund I got last year? Is it taxable? for additional information.
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