Hate to break out the I'm a math teacher thing, but a line actually has no width. A line is a one dimensional object, it only has length. This is all assuming we are using Euclidean geometry.
It's the practical application of drawing a line. For example, this is something we can recognize as a line:
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But we can also recognize a thicker line too, like the line made in the dark space here.
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Here's the pro tip. Don't be *that* guy.
And yes, Imothek's power should clearly read "line segment".
I do have one question too, what exactly is the standard procedure on determining a random model?
No standard procedure, so it's just whatever you and your opponent decide. Most of the time, a unit is small enough to roll a assign models spots on a d6 and roll. If no other dice are on hand (like a d20), then you could break bigger units up into groups of six, roll to see which group is hit while rolling another die to see which model is hit.
If you know this is something that will come up almost every game, it's good to bring appropriate dice for the occasion.