The equation is easily rearranged to give gross amounts going in either direction. It has been shown to you a hundred times or more.
P = k (T^4 - Tc^4) is exactly the same as
P = kT^4 - kTc^4
If all you want is an answer, then yes, they are the same...if you want to make a mathematical statement regarding a physical phenomenon, then no, they are not the same...in fact, they are quite different. And since there has never been a measurement of two way energy flow, the very idea of trying to substitute a mathematical statement that reflects every observation and measurement ever made for one that only applies in unobservable, unmeasurable, untestable mathematical models stinks of deliberate fraud.
The first arrangement emphasizes the temperature relationship, and shortens it because the components of k ( emissivity, S-B constant, and Area) only have to be written once.
The first arrangement is the actual equation used by the SB law..it describes a gross one way energy movement from cool to warm.
The second arrangement emphasizes the two gross flows that combine together to make a net flow.
The second statement is deliberate fraud...it makes a mathematical statement of a physical process that has never been observed or measured.
Both give the same answer. Neither arrangement is 'better' than the other.
The point of the equation is to describe a physical process..one describes an actual observable, measurable process..one is deliberate fraud which pretends to describe a physical process which has never been observed, or measured.
Follow through with your bastardized equation without violating SB's assumption that T1 is ALWAYS greater than T2
Just to put a period to the discussion, can you show me a two way Planck law by which to prove the two way version of the SB law? Answer: No.