As I have been wrestling with these ideas, the following is helping me clarify my own thoughts. (And thermodynamics was never my strongest suit, so I am not going to dive in with "answers").
Suppose I set up an airtrack with a pair of carts hooked together with a spring. If I set these moving, it is typical to think about the potential energy of the spring and the kinetic energy of each cart and use the language of Newtonian mechanics.
If I hook together many pairs of oxygen atoms and send them moving around the room, it is typical to think about the thermal energy of the various modes.
So, do we think about these differently because we are comparing
* large vs small? (if so, at what size should we switch? What about nanotechnology?)
* many vs few? (if so, how many is enough? What if I set up 10^4 pucks on an air table all connected by spring?)
* interacting vs non-interacting? (suppose I set up 1000 airtracks with pairs of carts; does it matter conceptually if they are weakly connected by springs vs each track moving on its own?)
Perhaps "thermal energy" is simply "potential energy and kinetic energy of objects that I am too lazy to deal with on an individual basis". :-/