Example 6 If the parabola $y=x^{2}+m x+2$ intersects the line segment $MN$ (including points $M$ and $N$) connecting points $M(0,1)$ and $N(2,3)$ at two distinct points, find the range of values for $m$.
Solution: The function expression of line segment $MN$ is $y=x+1$. Therefore, the original problem is equivalent to the equation $x^{2}+m x+2=x+1$ having two distinct real roots in the interval $[0,2]$. Organizing, we get
$$
\begin{array}{l}
x^{2}+(m-1) x+1=0 . \\
\text { Let } f(x)=x^{2}+(m-1) x+1 .
\end{array}
$$

To ensure that $f(x)=0$ has two distinct real roots in $[0,2]$, we need to consider the endpoints, the discriminant, and the axis of symmetry, which gives us
$$
\left\{\begin{array}{l}
\Delta=(m-1)^{2}-4>0, \\
f(0)=1>0, \\
f(2)=4+2(m-1)+1 \geqslant 0, \\
0<-\frac{m-1}{2}<2 .
\end{array}\right.
$$

Solving this, we get $-\frac{3}{2} \leqslant m<-1$.