Note: haven't picked up R+F yet, or any of the cash-grabbing DLC's yet. Maybe the definitions changed with them.
I now consider the "specialty" districts as the eight that provide bonus yields to internal trade routes, four food districts (campus, entertainment, holy site, theater square) and four production districts (commercial hub, industrial zone, encampment, harbor.)
I'm positive that entertainment districts (which provide internal trade route yield bonus) count as specialty districts even though they cannot house specialists.
I'm certain that aqueducts and neighborhoods don't count as specialty districts. This led to my original conclusion that all districts that are capped by population (to which aqueducts and neighborhoods do not) are the "specialty districts." However, at some point I think I confirmed that aerodromes and spaceports (that count towards the population cap) are not considered specialty districts on the basis that they did not add bonus yields to my CS trade routes when suzerein of Kumasi.