A line isn't really an area, being 1 dimensional instead of two-dimensional (when viewed on the 2-D grid).
The "Line" effect is new to 3.5, and uses a different rule than those for other "Area" effects.
Drivel.
We all know what a lightning bolt is supposed to look like, it is hardly one-dimensional. And there's nothing in the rules to suggest it behaves differently from other 'area' types.
Lightning Bolt
Evocation [Electricity]
Level: Sor/Wiz 3
Components: V, S, M
Casting Time: 1 standard action
Range: 120 ft.
Area: 120-ft. line
Duration: Instantaneous
Saving Throw: Reflex half
Spell Resistance: Yes
"Line" is not an effect. It is an area, regardless on the technical aspects of your grid. It is listed in the 'Area' section of the magic overview, almost immediately after "If the spell’s area only touches the near edge of a square, however, anything within that square is unaffected by the spell".
It then goes on to say:
A line-shaped spell shoots away from you in a line in the direction you designate. It starts from any corner of your square and extends to the limit of its range or until it strikes a barrier that blocks line of effect. A line-shaped spell affects all creatures in squares that the line passes through.
The diagram on PH pg. 176 that you reference so often doesn't show the four squares around the origin of the spell being affected. If your assumption was true, then all those squares would be part of the spell's area. The line needs to reach the far side of a square to affect it. I will grant that it affects four squares at intersections.