Naty I think you are confusing different phenomena.
There are two meanings to the term in Physics. 'blackbody radiation' and 'a black body'
Post#2 is correct, all bodies at the same temperature emit radiation (=light etc) of the same spectrum.
The term blackbody is used because the amount is zero at absolute zero. So the body viewed solely by its emitted light at absolute zero would appear black.
The amount of radiation varies from body to body and is the product of the surface area and a coefficient known as the emissivity. The emissivity for real material objects is always < 1 so these are called greybodies. A true blackbody has an emissivity of 1 and radiates the theoretical maximum amount of radiation.
Thermal equilibrium does not enter into the picture.
A body, grey or black, emits radiation as effieicently as it absobs it. The absorption coefficient is the same as the emission coefficient.
Any body continually emits radiation solely determined by its absolute temperature.
The same body continually absorbs radiation, solely determined by its surroundings.
The balance determines whether the body is undergoing radiative heating, cooling or is in radiative equilibrium. This is not the same as thermal equilibrium as the body may contain an internal heat source (eg a star)
On the other hand radiation by a black body is by definition zero at any temperature.
A black body is one which absorbs all incident radiation.
The coeficient appropriate in this case is called the reflection coefficient or albedo.
I think you will find Susskind's statement to be of the 'for all practical purposes' kind.