Felicia,

       You are right - the point of a cone is also called the vertex. Mathematics is a broad subject, with many parts, and some common words are used with distinct variations in different areas.

       The common 'idea' under both uses is that there is a 'point', and extreme value, when the object is built in 3-space.    So the corners of a tetrahedron and a cube (where three or more edges join up) are points in the spatial object.   Having several straight edges meeting is sufficient to produce a point, but not necessary.

       The point of the cone is also an extreme, but there is no standard way to see this with 'several edges' and no need to go that far. So a common word has taken on two different meanings in different parts of the math.

       Sometimes, people want to use 'vertices, edges, faces'  in order to work with a formula called Euler's Formula:   v-e+f=2  (number of vertices, number of edges, number of faces).  This is a particularly interesting idea, and has some rich connections.

       It does not, however, apply to circumstances where there are 'separate parts'.  That is where there are vertices like a cone, and edges like the bottom with no vertices and no edges connecting up to the vertex of the cone.

       The choice of words is about communication, and the context makes a difference to the meaning, in math, and in other parts of life.

Walter Whiteley.