I don´t know whether this kind of questions should be asked here or not.
I am trying to explain the concept of a homotopy class of simple closed curves (that is, the vertices of curve complexes) to someone who are not familiar with topological operations.
The first way is the one I like most. Define the simple closed curves as embedded image of circle, and then define two curves to be homotopic if and only if they can be connected by the operations given by moving out a bigon.
The second one is more nature for people who knows the definitions of homotopic maps. Just define the simple closed curves as maps from circles to the surface and define the homotopic class using the definitions of homotopic maps. But this explanation is not enough. This definition is exactly the definition of homotopy classes of oriented simple closed curves. What we really care is the embedded image, so we need to add that, if two curves with same image but different orientations, they are the same. But then, this explanation is not so neat for me now.
How can we explain the definition of homotopy classes of simple closed curves both neatly and naturally for people not working on geometry and topology.
You don't want to think about the curve as a parametrized object, it is just a curve on the surface. So you can do your second definition up to reparameterization of the circle (which is basically the same as up to orientation). Perhaps you like this view better or way of stating it better.