The difference is that the centre of mass is the weighted average of location with respect to mass, whereas the centre of gravity is the weighted average of location with respect to mass times local $g$. If $g$ cannot be assumed constant over the whole of the body (perhaps because the body is very tall), they might (and generally will) have different values.
I don't see an immediate connection with movement though.
I am tremendously confused about the exact difference between thermal energy and internal energy
Internal energy $E$ is all contained energy. And there are many ways to contain/store energy - one of them is thermally:
- In chemical bonds, when atoms join and form molecules
- In different phases of matter - ice at $0^\circ$ contains less internal energy than water at at $0^\circ$, because some energy has broken then bonds. Here a phase transition energy also called latent heat is the amount of energy needed to melt or freeze to do this phase change.
- As kinetic energy when particles move within a system
- As potential energies within the system when an object / objects is out of equilibrium (a book on a book-shelf wants to fall, or a compressed spring wants to jump out)
- As thermal energy, which is giving objects their temperature.
- Etc.
So no, internal energy and thermal energy is not at all the same. Internal energy is everything and includes thermal energy.
BUT when talking about ideal gases, their are no chemical interaction, no complex phase structure (because of no interaction), (usually) no significant potential energy. And the kinetic energy of each of the atoms is just the same as thermal energy or temperature. So here the inernal energy equals thermal energy - but only for an ideal gas.
Some websites loosely state that it is the "jiggliness" of the atoms/molecules that constitute the thermal energy of the system. What are they trying to say?!
Let's make this a bit clearer. Thermal energy is what we measure as temperature. So on the macroscale, if is felt as warmth/coldness.
But on the microscale, thermal energy is simply vibrating atoms. In a solid, they vibrate at their spot and more and more violently, as they are heated up. Soon they are heated so much that they vibrate so much that they rip themselves loose from the structure - and the materials is now melting.
This is why the word "jiggliness" is used. Thermal energy and temperature is in fact just microscale kinetic energy of the particles.
Best Answer
The terms are used interchangeably only outside of a scientific context, for example, in your kitchen, in the popular press or poor blogs, and even a few bad textbooks.
In a scientific context, you have it almost correct. Heat is the energy that enters or leaves a system on account of a difference in temperature (no work done).
Thermal energy is a component of the internal energy of the system. It is associated with properties that have a quadratic dependency on some parameter. It includes translational kinetic energy ($\frac{1}{2}mv^2$) as you point out, but it also includes rotational energy ($\frac{1}{2} I\omega^2$), and harmonic vibrational potential energy ($\frac{1}{2}kx^2$). Not included are things that do not have a quadratic dependence on energy. The most familiar perhaps is chemical binding energy (including the intermolecular binding energies in liquids and solids) but there can be others. The total of the thermal energy and the other energies is the internal energy.
The ideal gas particle has no internal structure, so it has no rotational energy, no vibration energy, and no chemical energy. So for that special case the thermal energy is equal to the internal energy, equal to the total kinetic energy of all of the particles in the gas.
There's some confusion about all this due in part to the fact that the basic concepts are introduced with respect to the ideal gas, but the distinction that occurs in applying the concepts to a real gas is often not made clear. A further complication is that it is a challenge to introduce the equipartition principle at a pedagogically early stage.