Two ways to calculate dissolved oxygen
The 21.0% of oxygen by volume in air is from dry air. Data for dry air from webpage of Columbia University's Department of Earth and Environmental Sciences.  

Note that the "% by volume"/100 for dry air also corresponds to the partial pressure in atmospheres. For air saturated with water at 25 C, the vapor pressure of water, according to Wikipedia, is 0.0313 atm. The second column is calculated by multiplying all dry air values by (1-0.0313)=0.9687. 

$$\newcommand{\d}[2]{#1.&amp;\hspace{-1em}#2}
\begin{array}{lrl}
\hline
\text{Gas} &amp;\hspace{-5em}\text{dry air}&amp;\ce{H2O}\text{ sat. air} \\
 &amp; \text{% by volume} &amp; \text{% by volume} \\
\hline
\ce{N2} &amp;  78.084    &amp; 75.640 \\
\ce{O2}  &amp; 20.947    &amp; 20.291 \\
\ce{H2O} &amp; 0.0      &amp;  3.13 \\
\ce{Ar}  &amp;  0.934    &amp;  0.905 \\
\ce{CO2} &amp; 0.035    &amp;  0.034 \\
\ce{Ne}  &amp;  0.001818 &amp;  0.001761 \\
\ce{He}  &amp;  0.000524 &amp; 0.000508 \\
\ce{CH4} &amp;  0.00017  &amp;  0.00016 \\
\ce{Kr}  &amp;  0.000114 &amp;  0.000110 \\
\ce{H2}  &amp;  0.000053 &amp; 0.000053 \\
\ce{N2O} &amp; 0.000031 &amp;  0.000030 \\
\ce{Xe}  &amp;  0.0000087&amp;  0.0000084 \\
\hline
\text{TOTAL}&amp; 100.0027187 &amp; 100.0026304 \\
\hline
\end{array}
$$

Now correcting the data from Warneck and Williams (evidently Warneck,  P.  and  Williams,  J.:  The  Atmospheric  Chemist’s  Companion:  Numerical  Data  for  Use  in  the  Atmospheric  Sciences, Springer Verlag, 2012) for the value of water saturated air

$\ce{O2} = \dfrac{20.291}{21}\times 8.47 = 8.18$ mg/L

This is much better agreement with the Benson (1980) data. 

The OP's value of 1.26e-3 Mol/L/atm for the Henry's constant for O2 in water at 25C was evidently take from Table 8.23 of Warneck and Williams. The value also has a list error bracket of +/- 0.02. 

$$\dfrac{0.02}{1.26}\times 100\text{%} = 1.6 \text{ %}$$

For the two dissolved oxygen values:

$$\dfrac{8.26-8.18}{8.26}\times100\text{ %} = -0.97 \text{ %}$$

so the values are within the error limits for Henry's constant.