You will need to check with the New Jersey courts or MVC however we believe that your problem lies in the fact that you are showing your international driving permit (IDP) instead of your valid Mexican driver's license to law enforcement.
The New Jersey Motor Vehicle Commission (MVC) states that as a non-citizen you may use your country's driver license as proof of driving experience if you have an International Driver License issued by your country or if your country is a member of the United Nations Convention on Road Traffic and your license is translated into English by a consulate or an MVC approved translator.
In the United States it is the valid license from Mexico that would need to be shown primarily to law enforcement and secondary would be the IDP since it translates your license for officers. The IDP by itself is not proper proof of a valid license from your country.
You mentioned the United Nations Convention which is intended to establish uniform reciprocal privileges on matters such as drivers' permits. The agreement requires the US and its states to allow a foreign driver admitted to the US to drive a private motor vehicle for one year on certain conditions.
[Let CarInsurance.com help you find affordable auto insurance now.] The NJ MVC notes that tourists from countries who are party to the Geneva Convention and the Regulation of Inter-American Motor Vehicle Traffic can also legally drive their own private cars bearing valid license plates/registration tags issued in their native country for a period not to exceed one year from their date of arrival.
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In addition to having a valid foreign driver license, NJ international visitors are advised to carry an International Driving Permit or attach an English translation to their country's national driver license prior to their arrival in the United States. Again they are advising you to carry the IDP but in addition to your valid foreign license.
The MVC though goes on to say that you are considered a visitor or tourist if you are visiting the United States for a period not exceeding one year and are not authorized to work or attend school in the country.
You may want to see if you can get charges or citations dismissed by showing the NJ courts your valid Mexican license. You may want to also find out for certain what documents the court would advise you to show law enforcement if you are stopped again so that you will not get ticketed or arrested again. | 
