Be aware that there is already a different function called “softmax”.
I named it (although others may also have done so) so I must take the blame for a neat but misleading name.
It should have been called softargmax: it provides a soft version of indicating which of several values is greatest in value, and it has several uses in so-called neural network algorithms. It is important to subtract off the maximum value from all the inputs before applying exp, for the reasons you explain.
See this summary: http://www.faqs.org/faqs/ai-faq/neural-nets/part2/section-12.html
It is the multiple-input generalization of the logistic: (1/(1+exp(-x))).

So to avoid confusion I suggest sticking to you name “soft maximum” for the function you describe, and avoiding the name “softmax”.

The choice of definition depends on what properties you want the soft maximum to have. If you want softmax(x, x) = x to hold, the factor of 2 you suggest works. But if you do that, then softmax(x, y) – max(x, y) no longer goes to zero as one of the arguments grows large.

log1p(exp(minimum-maximum)) + maximum

This avoids unnecessary precision loss when “maximum” is close to zero and minimum-maximum is less than -40 or so.

On another note, I am not sure I like how the “soft max” is always larger than the max. In particular, when x=y, shouldn’t any “max” function always return x?

log((exp(x) + exp(y))/2) is always between x and y, and reduces to x when x=y. (This is reminiscent of the “n-th root of the average of the n-th powers”, which approaches the max as n approaches infinity.)

In short, I might subtract log(2) from your formula.