For me the diagram needed to answer the question was difficult to draw and I did not make a video which would have illustrated the answer better because the soap film would have been easier to see.
I have made a complete circuit out of bare copper wire and within the circuit formed a soap film which represents a plane defined by the boundaries of the circuit.
In the left-hand image the two surfaces of the soap film, $S_1$ and $S_2$, are relatively easy to see?
There is no magnetic flux through surface $S_1$ and the magnetic flux through surface $S_2$ is $B \,A$ where $A$ is the area of the surface $S_2$ ie the area of the coil.
The total magnetic flux through the soap film is $BA$.
This is equivalent to the left hand diagram in the OP's question.
The right hand circuit which represents a solenoid with two turns was more difficult to photograph whilst at the same time showing the soap film clearly so I had to pull out the two turns into a helix with a much larger pitch than I really wanted.
This has resulted in the area of the soap film being increased and the magnetic field not being at right angles to the soap film.
What I write below would work for such a circuit but to make things easier assume that the two turns are close together as in a "normal" solenoid of two turns.
Again surface $S_1$ does not contribute to the magnetic flux.
Surface $S_2$ contributes $BA$ to the total magnetic flux passing through the whole surface of the soap film as does surface $S_3$.
The total magnetic flux through the soap film is $BA+BA=2BA$ which is $NBA$ with the number of turns $N=2$ again $A$ being the area of a coil.