First you said you understand the exact mathematical definition of these terms, that is not true! There are no defintions for these terms. A point, line and plane and distance are referred to as undefinable. The problems with Euclidean geometry is that it cannot be really completely considered pure math because these important concepts are not definied, ever. This is why the first 4 postulates cannot be proven since they rely on mathematical defintions of a point and line which we never defined. That is way geometry is not complete. As opposed to other fields of math where everything is on a solid foundation. David Hilbert tried to improve geometry based on this flaw but he never fully succedded, though he was able to get rid of pictures and diagrams which are used in geometry. You should understand that geometry is more intended for engineers and scientists so it really does not need such a strong foundation. Among mathematicians geometrical proves are not considered true proofs because they lack these definitons and important proofs.
Mathematicians think of a point as an ordered pair (x,y) in R^2 (in fact this can be generalized to higher dimensions). Where x and y are real numbers. That is it. Because it captures the applied meaning of a point. In a similar fasion mathematicans define distance as a metric,
d(x,y)=sqrt(x^2+y^2). All because it captures the idea of an applied meaning.