How do you determine the date a house was built?
All of this information will be available on most county (.gov) websites through title information or deeds.  They only care about when the house was titled, which as @bib has alluded to comes after a occupancy permit and a finished inspection.  

It does not matter when the house first started to get built.  The inspections will be done when it is done and will require everything to meet code when it is being inspected, not when the work was done.  So a house started in 1980 and finished in 1990 and inspected and titled in 1990 is "built" in 1990 for all intents and purposes (given that you do not own a historical home where this information might be important).