If I understand correctly, the HL is a (horizontal) plane on the eye level of the viewer.
It is not a plane but a line (H
ine), but yes, it is the same as the eye level.
Everything below (and above) converges to this plane.
Not quite, only horizontal
lines converge to the horizon
line - there is a reason for this linguistic correspondence
Other parallel lines, for instance of a roof, converge as well, but to a VP above or below the HL. But how to find them is advanced perspective, this class is only for horizontal lines.
But, this would mean that train tracks do actually not converge to the real horizon (the 'edge of the world' on a beach), but to a plane slightly above.
Definitely not so, the "edge of the world" IS the eye level.
If you don't trust geometry, try it out: let a friend take place between you and the visual "real" horizon. If you're of about the same height and the ground is flat, your head, his/her head and the "edge of the world" will be in one line.