In case of travelling by trains in London, please mind there is a difference between
London Overground and
National Rail network.
You can travel by London Overground or National Rail trains within London zones using a Contactless card or Oyster card.
London Overground fares are generally the same as Tube fares for similar journeys.
From my experience, travelling by
National Rail trains within London zones seems to be a bit more expensive sometimes, than travelling by the Tube.
According to the
tfl.gov.uk fares, the caps for Tube, DLR, Elizabeth line, London Overground and most National Rail trains within London zones are the same.