You have a misunderstanding here. Moving the wire in the flux is not the trick. The trick is to change the flux that goes through a wire loop. Moving a part of that wire loop so that the moving part cuts the flux lines is only one way to change the flux throug the loop. Other ways, such as rotating a magnet (a permanent one or electric) inside a generator also cause changing flux through a wire loop (=through the winding).
If we watch carefully your generator, we can see moving wires that cut the flux. Here is still no need for deeper explanations than you had in your 2 first images. But there are generator types which have no moving wires cutting the flux.
Another useful thing to know:
The emf in a generator or as well in the secondary winding of a transformer is not an interaction between the wire and the magnetic field. The emf exists as circular electric field around the changing flux. That's the induction. Metal wires conduct the electric field field for us to use. That conducting is known as influence. Having several turns in a winding is the way to have the emf connected several times in series.
This all probably sounds complex and distracting. In physics one can easily spot the following equation: F = qV x B, (=Magnetic Lorentz force) where F is the vector force on a moving charge q and B is the surrounding magnetic flux density vector. That equation easily explains the generators where a wire moves cutting the magnetig flux. But that equation is only another formulation for the much deeper fact. Actually the induction and other by magnetism caused forces are complex three party interactions where the parties are the charges that move and the space. My knowledge of relativistic electrophysics is practically worthless so it's no use to write any more some gorgeous sentences that I have heard. Except one: Electric and magnetic fields are simplified formulations of those interactions, simple but still accurate enough for everyday electrical engineering works and so rich in content that the special theory of relativity could be found from the basic equations of electric and magnetic fields.