Updated
What a twin-wall flue is
Two stainless steel pipes, one inside the other, with insulation packed between them. The insulation keeps the smoke hot as it rises, so the stove draws properly, and keeps the outer skin cool enough to pass safely through a ceiling or wall.
The two ways to run it
Internally: straight up from the stove, through the ceiling and the floor above, and out through the roof. It looks neat outside and keeps the flue warm, but it has to pass through the rooms above.
Externally: out through the wall behind the stove and up the outside of the house. Nothing changes upstairs, which suits a lot of houses — the flue is just visible on the outside wall.
Which one suits depends on the house, what is above the room, and where the flue can finish relative to windows, roof lines and neighbouring buildings. That is decided on a survey, not from a photo.
It is still Building Regulations work
A new flue is notifiable in the same way as any stove installation: the distances to anything that can burn, the height of the flue above the roof, the hearth, the carbon monoxide alarm and the notice plate all apply. Fitted by a HETAS registered installer, it is self-certified and you get the compliance certificate.