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 180-degrees in any triangle?)—but, later, non-Euclidian geometry was born from the assumption that one of his axioms was unjustified, thus, not necessarily true.