Usually NYS will suspend your registration, and maybe even your driving privileges, when they are notified by the Insurance Company that you lapsed, and they probably did that 10 years ago. The term of the suspension can only last for the number of days of the lapse, not to exceed the balance of the registration period on the plates. If that time period has already passed, (and there are no other issues with your driving record) then you should be able to "reapply" for a NYS driving privilege.
In NYS an officer has the right to request proof of insurance as part of a lawful car stop. If proof is not produced it creates a rebutable presumption that the vehicle was uninsured, that can be overcome with proof that it was insured. NYC also has a failure to produce a document violation.
A TCD is any sign, signal or marking that involves traffic. A speed limit sign is a type of TCD. A TCD ticket is a lesser violation carrying only 2 points. Even the lowest level speeding violation carries 3 points and the stigma of a speeding violation. A TCD can be considered a lesser included offense and we are seeing this type of ticket being issues in place of a speed more and more these days.
As a "resident" your status is no longer tourist and you need a local license. That fact that you have a NYS permit shows that you started that process, and only makes your situation more complex. The cited answer can be used as a means of arguing that you were following the rules, but most DMV judges rule that if you have resident status you need a license.