I think Michael, in the comment above, gave a convincing case for when to use "temperate" vs. "tempered". The answer by Matt also consolidates this point of "tempered" being associated to the result of an external action, since in all three examples the word "tempered" is followed by "by". Now where I disagree is about the conclusion to be drawn from this.

The delta function $\delta(x)$ is temperate because that's what it is. It is not like it originally existed in the form $\delta(x)+e^x$, and then some deus ex machina came and tampered with it (sorry couldn't resist) by subtracting the exponential, and finally made it into a tempered distribution.

I think Michael, in the comment above, gave a convincing explanation for when to use "temperate" vs. "tempered". The answer by Matt also consolidates this point of "tempered" being associated to the result of an external action, since in all three examples the word "tempered" is followed by "by". Now where I disagree is about the conclusion to be drawn from this.

The delta function $\delta(x)$ is temperate because that's what it is. It is not like it originally existed in the form $\delta(x)+e^x$, and then some deus ex machina came and tampered with it (sorry couldn't resist) by subtracting the exponential, and finally made it into a tempered distribution.