What are you taking a picture of? I.e. what is the subject, if it is the person then it seems to me it is sport; if it is the rock/cliff/mountain it's landscape. Think about it like this, if I started taking pictures of football grounds, that would be architecture but if I photographed football games, players in action then it would be sport.
As for the "is climbing a sport" debate then the word "sport" is one of those that has loads of meanings but I think bridge and chess are ruled out because they do not involve any physical skill whereas darts and snooker do. Some definitions require an element of competition but that does not need to be direct competition (e.g. things like long jump) and the best climbers are often trying to climb the hardest route, competing both with their own abilities and with others who are also trying to climb hard routes. There is also "Sport" in the huntin' shootin' fishin' sense of undertaking a physical activity for pleasure.