I'll try to respond to the OP.
I think this quote quite sufficient to understand temperature.
Mathematical Go: Chilling Gets the Last Point wrote:
[...]it is sufficient to think of temperature as a numeric estimate on the value of a move. The units are half the gote-value of a move in Japanese Go literature.
Later the idea, which I doubt is true, was spread around that Go players talked a lot about temperature. I think that is only true when Go players talk about theories such as those presented in the quoted monograph.
Like many concepts it has various realizations, depending on the intention.
For example
Let G be a game G := { L | R }, L and R also games.
Then define an operator, that we call cooling,
cool(G, t) = { cool(L, t) - t | cool(R, t) + t }
unless for some T < t, { cool(L, t) - t | cool(R, t) + t } is a number x (i.e. L - T <= R + T), then
cool(G, t) = x
Now G is said to have temperature T, mean(G) x and to freeze to cool(G, T).
Many other definition that allows for the following properties could be called temperature, even if they are not exactly the same, and I think you could be justified to call something temperature even if these properties are only usually correct. If one wished, then could be very precise about what is meant in each case.
Linearity: cool(G, t) + cool(H, t) = cool(G + H, t)
Order preserving: G >= H implies cool(G, t) >= cool(H, t)
mean(G + H) = mean(G) + mean(H)
temperature(G + H) <= max(temperature(G), temperature(H))
I doubt that these hold if we define
G := { f(L) | f(R) }
where f(x) is katago's score evaluation function, L and R as before. The most obvious violation is that f(x) isn't exact and will violate equalities and inequalities for that simple reason. Another problem is that while statements like G + H may work in form you usually can't actually add two 19x19 game together on a 19x19 board.
But it sure is similar in many ways, especially in form, to be useful.
I like that it is similar on form, I think that can be useful sometimes. If I were to suggest a less problematic definition that doesn't need to be similar in form then that would probably be
mean(G) := (f(L) + f(R)) / 2
temperature(G) := (f(L) - f(R)) / 2