Yes, this is one of the biggest misunderstandings of historic districts - there are 2 different types.
National Register Districts are established and determined to be important to American history, culture, architecture or archaeology by the National Park Service. Inclusion in a National Register Historic District is honorary and provides historic structures with limited protection from adverse effects by State and Federally related/funded projects.
Local Historic Districts are established by local city ordinance and are overseen by the Lafayette Historic Preservation Commission. Designation as a Local Historic District does not prevent owners from making changes to their properties; it simply guides them towards the most appropriate options.
If and when a property owner wishes to make changes to the exterior of their property they are required to complete an application for a Certificate of Appropriateness (COA) and present their proposed changes to the Historic Preservation Commission.