Pressure=Heat? Else, Heat=Radiation? A few days ago, I came up with this paradoxical conundrum:

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?

After a bit of thought, I could find some potential flaws/counter-logics to my proposal -

*

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

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

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

I think it'd be fair to admit though, the general "postulate" (if I am not wrong!?) that the supply of heat increases the randomness of the motion of particles in any medium.
This leads me to infer some interesting (but perhaps flawed) consequences -


*

*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?)


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

One possible flaw in this silly little thought-experiment of mine could be that our heat receptors work chemically. Meaning that electron transfer plays the pivotal role in the sensation of heat, not inter-molecular collision.

Then again, some form of radiant energy is necessarily absorbed to push the electrons out of atoms and cause the biochemical reaction that occurs everytime we feel heat. This radiant energy can't be anything other than radiant heat.

The reason I propose is simple - if radiant heat is not the "electron-pusher", something else is, then heat is not a factor to feeling hot (as that something else may push the electron at any other time, even when heat isn't supplied, and we may feel hot!) Otherwise, let's consider heat is the electron pusher, but some other form of it - convection or conduction. Conduction and convection both rely on molecules moving. I guess molecules can't push electrons out! So it must be radiant heat, I believe...
As a final summary of my paradox, here are the riddles that are forced, if the aforesaid inferences be true -

If the electron-push theory of heat reception be true, touching a hot metal wouldn't feel anything - Heat is conducted through metals, with molecules vibrating and making the heat flow. Molecules can't vibrate and displace electrons, right?! So we must not feel heat at all!
Else if molecular collision theory of heat reception be true, pressure and heat would be the same to feel - Cause you know, both are, after all, molecules striking your skin! What's the difference?

Or else -

Every kind of heat - conduction or convection, or even radiation - principally flows as a "stream" of radiant energy flowing through a medium or without any medium at all. The side effects of this flow of radiant energy, when it is flowing through a medium, include molecular vibration (conduction), random molecular motion (convection) and electron-donation on supply of energy.

I used the word "stream" of radiant energy, because the "side effects" (as I proposed) seems to me almost as if small channels dividing out of a stream...
That basically summarises my proposal at the conclusion, that what essentially transfers heat through a metal rod, when held near a furnace, is never the vibration of the particles of the rod. But the flow of radiant heat of the furnace that goes through the rod.
My question is, obviously, am I right? How much? Where am I wrong?
P.S.- I feel I'm wrong in a lot of places, so just maybe enlist where I'm right; that'll reduce your workload...
 A: 
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).

  
*
  
*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.


  
*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.


  
*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.

  
*
  
*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. 


  
*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 $W=P\Delta V$.  
Hope this helps.
A: Temperature of a gas is a macroscopic property that can be related to the average translational kinetic energy of its molecules.
Pressure = $\frac{F}{A}$ and $F = \frac{d\mathbf p}{dt}$
The average force in a given area of a container can be given by:
$F = n\frac{\Delta p_k}{\Delta t}$ where $n$ is the number of collisions in the time interval $\Delta t$, and $\Delta p_k$ is the average change of momentum in the normal direction to the wall for each collision.
So, it is not necessary that the temperature incresases to raise the pressure. If the volume is reduced, the number of collisions per time will increase. Even for the same average change of momentum of each collision (what is related to the kinetic energy and temperature).
About the way that our skin processes heat sensation, the difference between staying very close to a hot steel ingot (receiving mainly radiation) and touching it is huge. A worker can stand for some time in the first case, but in the second, without gloves, damage happens in a fraction of a second. So, I believe that radiation is not the only process of heat flow to the skin.
A: Increasing pressure of gas, doesn't always haye to be by decreasing volume. You can also keep the volume constant but, at the same time, increase the pressure of gas by supplying heat, thus increasing the temperature. That can be explained by p-T diagram, or more generally, p-V-T diagram. 
However, keep in mind there are sensible heat that can raise/decrease the temperature and latent heat that is needed to change the phase (example from water phase to gas phase)
A: Almost 2 months later, I seem to have found a fitting answer to my own question.

My question was aimed at how pressure is different from heat, if both involve vibration of paticles.

$ $
https://youtu.be/v3pYRn5j7oI
And there he is, Dr. Richard P. Feynman, telling us how :)
It turns out, that the extent of vibration is what differs across the two, heat and pressure. In the video, Mr. Feynman lucidly explains how heat is nothing but jiggling atoms excited by some input of energy. This is (perhaps) the precise point of difference between heat and pressure. Pressure is molecules gone rogue but only to a molecular level. But heat involves even the atoms going rogue.
I'd like to keep my answer short. Please comment if you spot any errors/have anything to object in what I had to say.
A: I am not an expert on the microscopic aspects, but I add the following.  Heat is associated with temperature and temperature is a measure of the average kinetic energy of the particles.  Pressure can do work and pressure is a measure of the average normal force per unit area of the particles on a surface.
A discussion of heat and work on the macroscopic level follows.
I think your example of the gas in the container needs clarification.  The result of heat or work (from pressure) is an increase in the internal energy of the gas. Heat and work are energy that is transferred across a system boundary; the gas has no heat and it has no work; it has internal energy that is changed by heat and or work done on the gas.
Here is a macroscopic discussion of the difference between Heat and Work. Heat and work are different, based on their definitions in thermodynamics. For a system, thermodynamics defines Heat as energy that crosses a system boundary- without mass transfer- solely due to a temperature difference. Work is energy that crosses a system boundary- without mass transfer- due to any intensive property difference other than temperature.  So work is a very broad concept, not just the force time distance concept of work used in mechanics.  [For example, mechanical work is a force acting through a distance; electrical work is electrical energy crossing the boundary of a system].   An "open" thermodynamic system is one with possible mass flow into/out of the system in addition to possible heat/work done on/by the system.
So pressure that moves a system boundary is associated with work and radiative heat transfer is associated with heat, and the effect of either on the system is a change in internal energy of the system.
For your example, if pressure from the surroundings acts quickly to compress the gas there is work done on the gas; there is little heat transfer between the surroundings and the gas for a rapid compression even if the surroundings and gas are at different initial temperatures (until later when the gas and the surroundings eventually reach the same temperature).  The temperature of the gas is increased by the change in internal energy of the gas from the work done on the gas. For an ideal gas, the internal energy is a function of temperature alone and by the ideal gas law an increase in pressure causes an increase in temperature.
