Pressure=Heat? Else, Heat=Radiation?

Let's say we have a gas enclosed in a container. Supplying heat to it will cause an increase in the (magnitude of?) vibration of gas particles in the medium. Increasing the pressure of the gas in the container (say by decreasing its volume) will increase the extent of vibration too. So what differs pressure from heat?

Let's take the simple case of an ideal gas where the internal energy depends only on temperature and the temperature is a measure of the average kinetic energy of the gas molecules. You can increase the temperature (what you call increase in vibration of the gas particles, but for an ideal gas it is really increasing translational velocities of the particles) by either heat transfer $Q$ to the gas, or by doing work $W$ compressing the gas (increasing the pressure) without heat transfer, or a combination of the two according to the first law

$$Delta U=Q-W$$

So "what differs increasing pressure from heat" is the former is energy transfer to the gas by work (compression), while the latter is energy transfer by heat (difference in temperature). The end result is the same (increased kinetic energy), but the mechanism for achieving the result is different (heat versus work).

1. Hotness is a sensation that a living body feels (by virtue of some receptor or something...)

Taking the skin as an example, hotness is a sensation that you feel when the thermorecptors in the dermis of the skin are exposed to elevated temperatures.

2. Hence the physical expression of the supply of heat may be the same as that of the supply (excess) pressure.

Although both heat and work (compression) can both increase temperature, it depends on what they are applied to. I've already discussed the case of a gas above. But if by "physical expression" you mean the feeling of hotness by the skin, energy transfer by heat would not be the same as by work in the form of compression (excess pressure) because the skin is not a gas.

To elevate the skin temperature by heat it needs to contact something higher than the normal temperature of the surface of the skin (about 33 C).The most common energy transfer by work would be friction work. Friction work is what warms your hands when you rub them together vigorously on a cold day.

As far as excess pressure goes, the type of work would be $pdV$ or pressure-volume work. That's the type of work that raises the temperature and pressure of a gas by compressing it. That type of work performed on the skin would not likely cause its temperature to rise. Unlike a gas which is highly compressible, liquids and solids are much less so. It takes extremely high pressures to raise the temperature of water a few degrees. Human skin is about 64% water.

3. The effect on the motion of gas molecules does not necessarily determine anything.

The effect of the motion of gas molecules is to increase or decrease its temperature.

1. When we feel hot, that sense must have something to do with the increase in the randomness of the motion of the "heat-receptor molecules" in the skin, and maybe their collision amongst themselves... (is this how they work?)

This is basically correct, as I already explained. But it's not the "randomness" of the motion per se, but an increase in the magnitude of the motions of the molecules that increases kinetic energy and the temperature that the thermorecptors sense and interpret as hotness.

2. The way "heat receptor molecules" can be excited (vibrated), is by supply of radiant energy, for that would be the best way to get them excited fast (because otherwise your only option left is to mechanically vibrate them)

This gets into the physiology of the how the thermorecptors function, which is something I'm not qualified to address. However, the fastest heat transfer means is by conduction (contact with a hot solid surface) not thermal radiation.

Mechanically vibrating the thermoreceptors if probably involved when you raise the skin temperature by friction work, the rubbing of hands together as discussed above. But compression (pdV) work is unlikely to do so.

This means a patch of radiant energy is integral and necessary to feeling hot

That is not correct. Heat transfer to the skin by conduction and convection can also make the skin hot, and do so faster than by thermal radiation.

UPDATE:

But just to be clear again, my question basically asks, what is the fundamental nature of heat?

The fundamental nature of heat is it is energy transfer between substances due solely to a temperature difference between the two.

Is it just energy radiated through vacuum, that sometimes collides with particles to produce vibrations, or is it the vibration as well?

The are three mechanisms of heat transfer: Conduction, convection and radiation. Energy transferred through a vacuum between objects of different temperature is radiant heat transfer.

Heat does not collide with particles. Nor is heat the actual vibration of particles. Heat is the transfer of the kinetic energy of the vibration, rotation and translation motion of particles from one substance to another substance. Heat is not the actual vibration of the particles themselves. That is called internal energy, specifically internal kinetic energy.

In the case of radiation through a vacuum, the kinetic energy of the higher temperature body is transferred to the lower temperature body by means of electromagnetic waves, or from the particle standpoint, by photons. The photons sent by the higher temperature body through the vacuum are absorbed by the particles (atoms, molecules, etc.) of the lower temperature body increasing their vibration, rotation and/or translational motion, depending on the energy of the photon.

In the case of conduction and convection, the energy is transferred by collisions between the higher and lower temperature particles.

How would you explain conduction as heat?

Heat conduction is the transfer of kinetic energy from the higher temperature body to the lower temperature body due to direct contact between the bodies. It differs from radiation since radiation does not require direct contact.

And what differs conduction/convection from pressure?

Conduction and convection are energy transfer by heat. Pressure by itself is only a force (i.e., force per unit area) and is not energy or energy transfer. However, pressure can transfer energy by causing a displacement of an object or substance. When a constant pressure is applied to a gas causing its volume to change, the source of that pressure transfers energy to that gas in the form of work $W=PDelta V$.

Hope this helps.