Well, what I think Euclid meant by "that which has no part" was a simple way of describing something which is completely and totally indivisible. A point has no length, width, volume, etc. If I'm not mistaken, I believe that's the definition of a point still used today. Similarly, "A line is the shortest distance between two points," is still the definition of a line. I'm not aware of any intention to model a pencil mark.
The link you pointed out says something about Euclid failing to realize that a certain few of his axioms were unjustified. I do not think it's referring to his definitions of a point and line. Unfortunately, I don't remember exactly which axiom it was that was unjustified (I believe it may have been something with there being 180degrees in any triangle?)—but, later, nonEuclidian geometry was born from the assumption that one of his axioms was unjustified, thus, not necessarily true.
