When two curves touch each other at a single point, are they called intersecting? When any two curves touch each other at a single point, are they called intersecting, or just called they are touching each other? Are these terms - intersecting curves and touching curves used interchangeably? Is there any difference between these terms? 
 A: "Interesecting" at $x$ just means $$f(x)=g(x)$$ "Touching" at $x$ additionally means $$f'(x)=g'(x)$$
A: An intersection between two curves is a point they both have in common. We say that two curves intersect if they have (at least) one point in common.
Two curves "touching" usually means they are intersecting and at the same time directions of the two curves at the point of intersection are exactly the same.
A: Let me clarify my above comment.
The discussion might be endless because there is no consensus on the definitions of "intersecting" and "touching".
One cannot give a definitive answer to the question raised until the definitions of these two words be clear and accepted by everybody.
Presently I doubt that the respective definitions be standardized, except in case of "intersection" in set theory.
In the set theory the "intersection" is well defined : That is the subset common to the two sets considered. https://en.wikipedia.org/wiki/Intersection_(set_theory) .
If this definition in set theory was extended to usual geometry the intersection of two curves would be the common part of the curves whatever the overall configuration and whatever the common part be one point only or many. But everybody can or not support this view.
I am not qualified for the standardization of mathematical vocabulary. Nevertheless I am allowed to give my own opinion which is :


*

*For geometry, generalize the definition of "intersection" well established in the set theory.

*And for geometry, standardize the definitions of some sub-cases such as "Crossing  intersection", "Touching intersection", etc.
