You already have all the ingredients for an answer.
Internal Energy ($U$) is a property of the system (e.g. the gas has a definite energy) while heat ($Q$) is transfer of energy (e.g. if i put two systems with different temperature together they will exchange some energy and we call that heat).
Everything is linked together via the first principle of TD
$U=Q-W$ where $W$ is the work applied to the system or done by the system (the signs are such that we are using the "point of view" of the system, so that $Q>0$ if the system is receiving heat, and $W>0$ if the system is doing work).
Thus if I want to decrease (increase) a system's energy I can:
a-have it do some work
b-extract some energy from it, for example by placing a colder system near it: the system will transfer some of its energy to the other system and that is heat.
Now, this is very general.
As you said particles in the system move and thus part of the internal energy is given by the particles' kinetic energy. Thermodynamics tells us that the average kinetic energy depends on the temperature and is therefore dubbed "Thermal Energy". Thermal Energy is of course energy, it is a quantity associated to a system at a given temperature.
In the example above in which we extracted energy from a system putting it near a colder one, what happens eventually is that our system will be now colder thus it will have less kinetic energy and thus less internal energy. The missing energy has been transferred to the other system as heat.
As you said, a system does not have a "heat content", it only has "energy content" BUT sometimes terms such as heat content or heat reservoir are used to describe the amount of heat one can extract from a system, i.e. how much energy can I transfer from the system to somewhere else.