Tesla coils generally use near-field energy transfer. But ...

The page cited is sloppy in its wording and it would better to say "near field" where it says "RF". But, (again) ...

The answer is necessarily not black and white.
"Near field" has a precise meaning BUT obtaining ONLY near field coupling in a given case is not certain.

There is a gradual transition from NF to FF and a boundary region where both can occur. Near Field essentially occurs where certain geometry dependant terms form a major or significant part in describing interactions between transmit and receive structures - these relate to cyclical transfer of energy between the antenna I and V and adjacent electrostatic and magnetic fields. As distance increases these terms become less significant until they can be ignored.

You'll always get "some of both" with "some" varying as you move away from the antenna.

(1) At distances of well under a wavelength there is substantial interaction between the electric and magnetic fields produced by the aerial and current and voltage in the antenna. Energy is transferred to and fro between fields and aerial throughout with losses caused by non idealities but no energy loss due to energy "leaving" the aerial structure. This close in zone is termed the "reactive zone" where power may be absorbed by a tuned load which has voltage and current induced in it and which then dissipates energy (ie has a resistive component). Coupling involving power transfer is magnetic.

(3) [note number] "RF communications or energy transfer occur at distances beyond several wavelengths form the"antenna" structure. Here the ratio if electric and magnetic coupling have "settled down" and any energy present is not coupled to the structure I & V so is "lost", whether "received" or not. One way of viewing this one is that the two aerials are geometrically distance and secondary terms which account for the filed coupling and which have a strong distance dependent component have become insignificant - the field has become essentially homogeneous over lengths of the order of the receiving antenna.

(2) At distances past about half a wavelength the "second order" terms on which pure NFC coupling depends start to get small and the field starts to become homogeneous. This is termed the "Fresnel zone" (the guy has his name all over) and there is a degree of non ideality in field coupling to the antenna.

_

This wikipedia page on near and far field does a better than usual job of commenting.

Their summary section says, in part:

• The near-field is remarkable for reproducing classical electromagnetic induction and electric charge effects on the EM field, which effects "die-out" with increasing distance from the antenna (with magnetic field strength proportional to the inverse-cube of the distance and electric field strength proportional to inverse-square of distance), far more rapidly than do the classical radiated EM far-field (E and B fields proportional simply to inverse-distance). Typically near-field effects are not important farther away than a few wavelengths of the antenna. Far near-field effects also involve energy transfer effects which couple directly to receivers near the antenna, affecting the power output of the transmitter if they do couple, but not otherwise. In a sense, the near-field offers energy which is available to a receiver only if the energy is tapped, and this is sensed by the transmitter by means of answering electromagnetic near-fields emanating from the receiver. Again, this is the same principle that applies in induction coupled devices, such as a transformer which draws more power at the primary circuit, if power is drawn from the secondary circuit. This is different with the far-field, which constantly draws the same energy from the transmitter, whether it is immediately received, or not.

There are two reasons why your earlier question wasn't about radio. The first is, that radio officially goes from 3kHz to 300GHz. The second is, that a transformer is based on a different principle than radio waves. That second reason is what's your question is about: a transformer is based on electromagnetism, radio waves are based on electromagnetic radiation.

Understanding on this topic is really hard, and exists for many people on a lot assumptions. I'll try to give an easy explanation for a layman, for which you'll have to accept some more assumptions than for the detailed explanation below.

Layman explanation

As you know, a magnetic field means that some materials like metals are attracted by others. One can generate a magnetic field by letting an alternating current flow through a wire or coil. That is what happens in the primary coil of a transformer. The other way around, a change in a magnetic field will generate a current in a coil - that's what happens in the secondary coil. These properties of magnetic fields and current are called electromagnetic induction.

Electromagnetic radiation is a particular form of the electromagnetic field. In electromagnetic radiation, the magnetic field will create an electric field (just assume that), but further away from the conductor that began with making the electromagnetic field. The electric field will create a magnetic field, even further away, and so on. It just goes on and on, due to specific properties of the field. That's the key to electromagnetic radiation.

When you are testing with a transformer, the secondary coil exists inside one wavelength of the wave that is produced. This means that the current in the secondary coil does not exist because of electromagnetic radiation, but because of electromagnetic induction: the fields don't create each other.

You can only prove the existence of electromagnetic radiation by transporting waves over more than one wavelength - only then, you can be sure the fields create each other.

Detailed explanation

There is some confusion here, and the cause of that is that the theoretical principle behind radio waves, and radio frequency, don't necessarily go together. Take a look at the Radio Wikipedia:

Radio is the wireless transmission of signals through free space by electromagnetic radiation of a frequency significantly below that of visible light, in the radio frequency range, from about 30 kHz to 300 GHz. These waves are called radio waves. Electromagnetic radiation travels by means of oscillating electromagnetic fields that pass through the air and the vacuum of space.

^{Note: I believe the 30kHz minimum should be 3kHz (reference: here and here)}

You can see that there might be other waves, based on the same principle and working the same way, with a frequency <3kHz or >300GHz, that are just therefore not part of "Radio". Those waves aren't radio waves and they aren't in the RF spectrum, but they are just the same, when you forget about the frequency.

But there's more! Radio waves are electromagnetic radiation. Electromagnetic radiation contains of two components, one electrical and one magnetic. These components create each other, as said above. The red magnetic field creates a blue electric field, which creates the next magnetic field, and so on.

From the Electromagnetic radiation Wikipedia:

Electromagnetic radiation is a particular form of the more general electromagnetic field (EM field), which is produced by moving charges. Electromagnetic radiation is associated with EM fields that are far enough away from the moving charges that produced them that absorption of the EM radiation no longer affects the behaviour of these moving charges.

What we were trying to do in your earlier question was really just picking up the weak magnetic field, because that's what a secondary coil does.

I guess you're now wondering: but does a transformer do electromagnetic radiation, or is it just a magnetic field? Let's have a look, with the Electromagnetic radiation Wikipedia:

... the electric and magnetic fields in EMR¹ exist in a constant ratio of strengths to each other, and also to be found in phase ...

^{1: electromagnetic radiation, compared to the electromagnetic field - note by author}

Think about the transformer. A magnetic field is generated when the current changes. Let's say we have a pure sine as the current, \$I(t)=sin(t)\cdot{}c\$. We can get the change of the current on a specific moment by taking the derivative of that sinus, which is the cosine, so: \$B(t)=cos(t)\cdot{}c\$. Now have a look at the functions \$I(t)\$ and \$B(t)\$, which should exist in "a constant ratio of strengths to each other" and in phase.

^{Note: the constant \$c\$ is because the formulas depend on other things as well, that are irrelevant now and constant in a specific situation}

You can already see those functions aren't in phase. They aren't in a constant ratio to each other either. You can see that by plotting \$f(t)=\frac{sin(t)}{cos(t)}=tan(t)\$:

So no, a transformer does not radiate electromagnetic radiation. The waves aren't in a constant ratio of strength to each other, neither are they in phase. The tests you did with a transformer in your earlier question, were just based on a magnetic field.

This difference between picking up a magnetic field and magnetic radiation is known as the difference between near and far field.

Summary

There are two main reasons why your experiments weren't about radio. The first is that it just was the wrong frequency. The second is that a coil with an AC current does not provide electromagnetic radiation.

Reference

• Wikipedia: Radio, Radio frequency, Electromagnetic radiation, Electromagnetic field, Near and far field
• Industry Canada, GL-01
• Your earlier question here
• Plots were made with Wolfram|Alpha