electromagnetic-radiationhuygens-principleopticswaves
How and why does the Huygens principle really work? I mean, does it always work?
The Huygens principle:
Every point on a wave-front may be considered a source of secondary spherical wavelets which spread out in the forward direction at the speed of light. The new wave-front is the tangential surface to all of these secondary wavelets.
For the Huygens principle to be true, light must emit light. Can light emit light?
One thing that the Huygens principle means, and I think it is this that you're getting at, is that the propagation of a wavefront is independent of where the light actually came from.
So yes, when a ray of light inside a room reflects off the walls, the walls act as a new source. If the original light source didn't exist, but the walls emitted a wavefront with exactly the same amplitude and phase as before, then you wouldn't be able to tell the difference.
In the figure that you titled as "correct way", indeed everything is correct. You say that you have a problem, "these secondary wavelets are all formed at different times". Yes, the wavelets appear at different times, and this is why the reflected wave-front is inclined. Look at the picture below.
By the time the leftmost ray reaches the point $A$, on the wave-front (dark green) the other two rays are far from the reflecting surface, points $B$, respectively $C$.
By the time the point $B$ reaches the surface, point $B_1$, the surface of the wavelet formed around $A$ touches the point $A_1$.
Finally, by the time the point $C$ reaches the surface, the wavelet around $A$ touches the point $A_2$, and the wavelet around $B_1$ touches the point $B_2$. The reflected wave front is colored red.
Best Answer
The key phrase is "may be considered". That is, Huygens principle, as a mathematical procedure, gives (approximately) the right answer. The principle does not say that light is actually being generated at each point on the wavefront.
Another way to look at it: the light at a phase front already exists there. There is no need to generate it all over again.
It's interesting to note that if the vacuum were a "perfect" polarizable material, then the incident light would polarize every point in space, and each point would scatter light into spherical waves. I don't think that there's any physical justification for the picture, and it might lead to dangerously wrong conclusions, but it might help give you a grasp of how the mathematics works.