Why is it considered acid rain with pH <5.6?
The $\mathrm{pH}$ of pure water (rain as well as distilled water) in equilibrium with the atmosphere ($p_{\ce{CO2}}= 10^{-3.5}\ \mathrm{atm}$) can be calculated as follows.

$$[\ce{H2CO3^*}]=K_\mathrm H\cdot p_{\ce{CO2}}$$

where $[\ce{H2CO3^*}]$ is the total analytical concentration of dissolved $\ce{CO2}$, i.e. $[\ce{H2CO3^*}]=[\ce{CO2(aq)}]+[\ce{H2CO3}]$, and
$K_\mathrm H= 3.39\times10^{-2}\ \mathrm{mol\ l^{-1}\ atm^{-1}}$ is Henry's law constant for $\ce{CO2}$.

$$\begin{align}
\log[\ce{H2CO3^*}]&amp;=\log K_\mathrm H+\log p_{\ce{CO2}}\\
&amp;=-1.5-3.5\\
&amp;=-5.0
\end{align}$$

The commonly used first acid dissociation constant of carbonic acid $\mathrm pK_{\mathrm a1}=6.3$ (at $25\ \mathrm{^\circ C}$) actually is a composite constant that includes both the hydration reaction 
$$\ce{H2O + CO2(aq) &lt;=&gt; H2CO3}$$
and the protolysis of true $\ce{H2CO3}$
$$\ce{H2CO3 &lt;=&gt; H+ + HCO3-}$$
For a weak acid
$$\begin{align}
\log[\ce{H+}]&amp;\approx\frac12\left(\log K_\mathrm a+\log[\ce{H2CO3^*}]\right)\\
&amp;=\frac12\left(-6.3-5.0\right)\\
&amp;=-5.65\\
\mathrm{pH}&amp;=5.65
\end{align}$$

Thus, pure rain in equilibrium with the atmosphere has about $\mathrm{pH}=5.65$. Any acid rain with lower $\mathrm{pH}$ would be caused by additional acids.