To answer your questions. Whenever a subdivision is made it has lots numbered 1, 2, 3, 4 etc. So as you can imagine, when you have 500 subdivisions there are 500 lot #1s. To ensure that every lot has a specific designation number the lots in subdivisions are systematically undergoing the reregistration process where they are given a specific designation number these are referred to as parcels. Not all subdivisions have yet been integrated into the national grid (parcel) system but they will be over time.
In order to ensure you lot number and parcel number are correctly matched you would hire a title investigator who will pull up the necessary survey maps and other paperwork to ensure that the info meshes correctly.
No, I doubt that whoever you hired to pay your property taxes is qualified and capable of providing you with the quantifiable evidence you are requesting - hire a lawyer.
If you don't trust your payment agent and you are unsure, as I have said before, hire a lawyer. That is what lawyers do and they are qualified to provide you with the correct advise.
I have been in the real estate business for 4 decades I hire attorneys to handle such matters as there is the confidence that they are qualified to correctly advise you and can ensure that searches are thorough etc. You have time to deal with a lawyer - because it is important and valuable.