From the pics I could find, both an arbour and a gazebo are usually not attached to a house???
Right. Usually freestanding. If it's attached to the house, out west here we say "covered patio" for both solid roof and pergola style.
Pergola did sound Aussie. I used to watch the home improvement show on BBC and they were forever building free standing pergolas with water features.
Anyway, it worked in the story.
Artemis