The following ticket types are fine to use for 2FOR1 offers:
National Rail tickets - purchased from a rail station, including: Advance, Anytime, Off-Peak, Gatwick/Stansted Express
Travelcards - only when purchased from a National Rail station.
Print @ Home/Mobile - showing the National Rail logo only Nrlogo
Season tickets with Photocards - including those with TfL Travelcard add-ons
Annual Gold Cards
Annual Oyster card WITH Gold Record card