Why the name for isoelastic utility? If $u(x)=\frac{x^{1-\mu}}{1-\mu}$, then the elasticity defined as $\frac{\partial u}{\partial x}\frac{x}{u}=1-\mu$.

However, when we define $u(x)=\frac{x^{1-\mu}-1}{1-\mu}$ (usual def. for isoelastic utility), we have $\frac{\partial u}{\partial x}\frac{x}{u}=(1-\mu)\frac{x^{1-\mu}}{x^{1-\mu}-1}$. And this doesn't seem to give a constant elasticity. So, why do we call this utility isoelastic?

Any help would be appreciated.
Iso-elastic utility is defined as a function $U(w)$ where for all $k \gt 0$ you have $$U(kw)=f(k)U(w)+g(k)$$ for some functions $f(k)$ and $g(k)$ independent of $w$, i.e. an affine transformation of the original utility function

This is iso- because this kind of utility function leads to identical (or at least proportionate) decisions: it means that the optimal risk/reward behaviour at given level of wealth $w$ is also optimal for another level of wealth $kw$ if you multiply all the original investments by $k$.  For example, if at some wealth level you would put a third of it into a particular risky investment and two-thirds into a particular safe investment, then with an iso-elastic utility function you would do the same with at any wealth level   

Both your formulations are iso-elastic in this sense, since they only differ by a constant $\frac{1}{1-\mu}$ and so lead to identical distribution decisions.  The difference between them is that the first has $u(0)=0$ and $u(1)=\frac{1}{1-\mu}$ while the second has $u(1)=0$, and since that difference has no real-life implications it might be unreasonable to claim the two utility functions had different properties  

Those who use the second formulation often associate it with a logarithmic utility function as the limit function as $\mu \to 1$.  This limit cannot be done without subtracting the $\frac{1}{1-\mu}$ (no longer a constant, since $\mu$ is changing), so they take the second formulation for arithmetical convenience