Ashley,

This is a common query, so let me give an answer that has appeared many times:

The answer you seek depends on the context / purpose / generalizations you have in mind.

Often people are interested in vertices, edges, faces of a solid as part of an exploration of Euler's formula (v-e+f = 2 for surfaces without holes). For that purpose, you want to say all surfaces which enclose the interior are identified as faces, and we also want the 'faces' to be like polygons or 'discs'. So think of how you would CONSTRUCT the cylinder and cone from (flat) paper, cutting, and sticking together. A cylinder might be made from a rectangle of paper, and two circles - 3 faces. You would stick them together with three pieces of tape (one on the top, one on the bottom, and one up the side from top to bottom to close the rectangle into a tube) so 3 edges. You have two points where a couple of pieces of tape end (actually three ends of pieces of tape at each of these points) one on the top rim, one on the bottom, so
2 vertices: v-e+f = 2-3-3=2!

For a cone, you need two pieces of paper, and only two pieces of tape, with one two points where pieces of tape ends: v-e+f = 2-2+2 = 2!

You might also want to identify 'faces' because you are measuring the surface area. The same cutting up used above gives simple pieces whose area is known, and since all parts are covered, you get the whole area.

You might be wondering about how many ways it can 'sit' on the table. The hedra in polyhedra is 'seat' so it is reasonable to ask about how many ways to 'sit'. For this you want flat faces, and the rounded cylinder and the rounded cone will cause confusion, so people tend to insist on flat faces for this context!

It often happens in math that the definition you really want depends on the context (what problem are you trying to solve), as well as what generalization might you want to make. So different books and web sites will give distinct answers depending on the context.

Walter