Waldo said:
First, I'd like to cool it down. I had bad day and I was feeling argumentive. I apologize for that. But... 
No problem. But I do think you are making this far more complicated than it needs to be.
You keep saying this, but I don't see where this is actually stated. The diagram is the only place that contains "Or touches" and it doesn't say, "The origin point isn't included" nor does the text, and it doesn't show the end of the line affecting squares beyond its reach, even though the tip of the arrow reaches the other end of the last square.
The line is only 120' long. Can't affect anything beyond that range (as stated in range definition on page 174-175 of the PHB).
If we assume it affects any square it touches, ignoring the rest of the area rules, that would include the two squares to the side of the origin, and the square at the tip of arrow (and the caster's square, but that's obviously not the case).
PHB, page 175, last paragraph: "A line spell
shoots away from you in a line in the direction you designate" Hitting the two squares to the side of the origin, or the casters square, would not be "away from you", it would be towards you or to your side.
A line begins at an intersection, like all spells. You could cast it from any body part or anywhere else and the intersection is all that's important. Your hand probably isn't at the intersection, it's in the square somewhere. It's also a 5' wide path of destruction at your fingertips. All random things we can't really determine from rules. Either way, you're not acknowledging the fact that the line touches those squares and doesn't affect them. You're using the diagram both ways. It's contradictory evidence.
I acknowledged that "fact" in my last post. I just don't think it's contradictory evidence, because the line extends
away from the point of origin.
In addition, the range of Lightning Bolt, as well and the area, is 120 feet. "If any portion of the spell’s area would extend beyond this range, that area is wasted", so those squares at the end are out of range and aren't affected.
And that doesn't contradict my position, it's simply a different rule that limits the area. I'm concerned about the area within the range, not any square that might theoretically extend beyond it.
In fact, this supports my case. range of the spell is exactly 120', starting at one of the corners of the casters square, and can't extend beyond that range,
in either direction, which would preclude if from hitting the caster's square, or the adjacent squares.
Also, try looking at it another way:
By your interpretation, a line that extends exactly from diagonal to diagonal out to the limit of it's range would only affect 16 squares, missing everything in eveyr ajacent square.. By mine, it would afect 46 squares.
By both our definitions, a line that extends straight out along the gridline would affect 48 squares.
Do you really think it is intended that a diagonal line should only affect 1/3 the area of a straight line?
I would tend to think the interpretation that gives nearly equal area of effect no matter which direction you shoot the line in is most likely to be correct.