Step 1: Name the points and the ratio.
Let $A(1,4,6)$ and $B(5,2,10)$, and suppose the dividing point $(a,8,-2)$ splits $AB$ in the ratio $m:n$, which we write as $\lambda:1$ with $\lambda=\dfrac{m}{n}$.
Step 2: Use the $y$ coordinate to find $\lambda$.
By the section formula, $8=\dfrac{\lambda(2)+1(4)}{\lambda+1}$.
Step 3: Solve for $\lambda$.
Cross multiplying, $8\lambda+8=2\lambda+4$, so $6\lambda=-4$ and $\lambda=-\dfrac{2}{3}$. Hence $\dfrac{m}{n}=-\dfrac{2}{3}$ and $\dfrac{2m}{n}=-\dfrac{4}{3}$.
Step 4: Find $a$ from the $x$ coordinate.
Using $a=\dfrac{\lambda(5)+1(1)}{\lambda+1}$ with $\lambda=-\dfrac{2}{3}$, the numerator is $-\dfrac{10}{3}+1=-\dfrac{7}{3}$ and the denominator is $-\dfrac{2}{3}+1=\dfrac{1}{3}$, so $a=-7$.
Step 5: Quick check with the $z$ coordinate.
$z=\dfrac{\lambda(10)+6}{\lambda+1}=\dfrac{-\frac{20}{3}+6}{\frac13}=\dfrac{-\frac{2}{3}}{\frac13}=-2$, which matches the given $-2$, so the ratio is correct.
Step 6: Evaluate the asked expression.
$\dfrac{2m}{n}-\dfrac{a}{3}=-\dfrac{4}{3}-\dfrac{-7}{3}=-\dfrac{4}{3}+\dfrac{7}{3}=\dfrac{3}{3}=1$.
\[ \boxed{1} \]