You're likely to find as many definitions of "rig" as there are animation/modeling programs. I've even seen "rigs" for creating rigs. Only my opinion: I generally don't think of a rig as including the skeleton. It's a means to generate animation data for a given skeleton, intended to be simpler and/or more intuitive than rotating individual bones/joins/frames.

Are the artists meant to use them as an asset in their game or render? Or are they just to be used by the animator, and then the modeller will take the skeleton and skin the actual character mesh?

It's unlikely that a rig would be considered as an asset for the game itself. It's a means to create animation (asset) data, probably stored with the project for creating the app, but not likely to be distributed with it.

With regard to the entire process, it's recursive. Overall, the size of the mesh, number of bones, number of bone-weights, number of animation steps, etc., "should" be as small as will produce satisfactory results. A common sequence is: 1) create the character (mesh) 2) skin the character (add/adjust skeletal joints/bone-weights) 3) animate the character. During the animation phase, e.g., a "required" bending of an elbow results in an ugly crease. A decision is made whether to live with it, or change the "requirement," or change the bone-weights, or add another bone, or add another bone weight, or add a few more vertices, etc. Each of those decisions has a penalty.

I think of a rig for animation like wrapper code for libraries... they are a set of controls to carry out complex actions more simply, without needing to work exclusively with the lower-level implementation. Only instead of doing something with code it's doing character movements. It could be something as complex as making an entire character turn around realistically with one "lever", with waist and shoulder movement going along as secondary actions, or an inverse kinematics control (common in most 3D animation programs). It lets you move one end of a bone structure of a series of joints while moving all the joints along with it.