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Your understanding is correct.
I don’t think you’ll find a clear direct statement from the state about what is and is not deductible. The New York tax return goes about it in a roundabout way, first adding up all the federal deductions, including state income tax (but without the $10,000 cap), then subtracting out the state income tax later. See Form IT-196 and the instructions for that form. State income tax is entered on line 5, then subtracted on line 41.