To start with there exists two frameworks of mathematically describing radiation. The classical framework with Maxwell's electrodynamics, and the quantum framework, with the quantum equations ( Dirac, Klein Gordon , started with Schrodinger) and with Quantum ElecroDynamics (QED).
1."Quantum radiation": radiation during a quantum state change, one photon is emitted.
This belongs to the quantum mechanical framework and can be described by solutions of the QM equations as transitions in energy states.
- "Classical radiation": radiation of accelerated charge, a lot of photons are emitted.
This belongs to the classical framework, which can be shown to emerge from the quantum mechanical.
The Quantum mechanical framework was needed and therefore formulated because of data that could not be predicted by classical electromagnetism : spectra of atoms, photoelectric effect, black body radiation. QED is the underlying framework for all electricity and magnetism phenomena.
What is (or is there) the essential difference between the two form of radiation process in more advanced physics like quantum field theory?
The classical electrodynamics emerges from QFT , similar to how mathematically thermodynamics emerges from statistical mechanics. The classical theory is useful in predicting macroscopic behavior of matter to electricity.
How to view the second form in a quantum way? Like see the accelerated charge as a series of quantum state changes and each of which may emit a single photon like the first form does - then in this way the second forms is just a classical view of a series quantum state changes.
The quantum form is QED, it is always a particle interaction. The emission of radiation from a charged particle in the table of elementary particles is given by the calculation of feynman diagrams, derived from QED, see the diagrams here, for example.. In this link there is a description in term of photons.
Examples of radiation process forms that between the two "extreme forms"? When talking about light detectors or LEDs,
This is basic quantum mechanics, transition between bound states by photon emmission.
forms one is used. When talking about antenna, forms two is used. No form in between?
This is classical electrodynamics, dealing with zillions of quantum particles, the electrons. They build up the fields but it is much simpler to deal with the classical electrodynamics.
What is the radiation form for thermal radiation?
It is black body radiation, one of the main proofs that quantum mechanics is needed even for bulk processes.