Subexponential growing functions

What is the most common definition of a subexponential growing function ? It seems there are different notions in literature.

I'd say that the most common one I see (and I should note that this is generally in the world of complexity theory/algorithm analysis) is a definition that distinguishes a subexponential function $f$ from exponential functions above it and polynomial functions below it, and moreover distinguishes it from 'polynomially diminished' exponential functions like $x^{-2}e^x$; that is, $f$ is subexponential means both that $\lim_{x\rightarrow\infty}f(x)x^{-\alpha}=\infty$ for all $\alpha$, and $\lim_{x\rightarrow\infty}(\log f(x))/x=0$ (which is a stronger condition than requiring that $\lim_{x\rightarrow\infty}f(x)\beta^{-x} = 0$ for all $\beta\gt1$).