Tool that modifies number of vertices but not the feature itself

Those segments between vertices look like straight line but they are not. They're Bezier or circular arcs. This behaviour is all over ArcGIS system. For example take a look at these articles:

• http://support.esri.com/en/knowledgebase/techarticles/detail/18943

• http://support.esri.com/en/knowledgebase/techarticles/detail/18944

To solve this situation after using the "Intersect tool" use simplify line

Or maybe see How to Identify and Delete duplicate vertices?

CLIP: In clip tool, it can’t clip features as arc. It clip feature clip as a straight line. Due to straight line vertices are increased.
NOTE: if you want to clip polygon without increasing vertices then you should clip it by reference of line feature class by using “split polygons tool”.

INTERSECT: When you use intersect tool on line feature with some feature (Line or polygon), it increase one vertex, because it shows from which vertex your line had increased.