In SR consider two time like separated events -
In some frame \begin{equation}ds^2= dt^2 - dx^2\end{equation}

In a frame where the events occur at the same place ( rest frame; $dx' =0$) **then according to what I know proper time is the time elapsed in that frame i.e. $d\tau=dt'$**.

Hence \begin{equation}ds'^2 = d\tau^2 =dt'^2\end{equation} ( since $d\tau =dt'$ in that frame)and since the interval is invariant; 
\begin{equation}
d\tau^2=dt'^2= dt^2 - dx^2.\end{equation}

Consider the same event in GR, in some frame ( coordinate system) \begin{equation}ds^2= g_{00}dt^2 - g_{11}dx^2....\end{equation}
In the frame ( coordinate system) where the events occur at the same place ( $dx'^2=0$) **then according to what I know proper time is the time elapsed in that frame i.e. $d\tau=dt'$**,  and we should have, \begin{equation}
ds'^2= g_{00}dt'^2 = g_{00}d\tau^2\end{equation} ( by the same analogy as in SR, $d\tau =dt'$) and since the interval is invariant, we should have, \begin{equation}
g_{00}d\tau^2 = g_{00}dt^2 - g_{11}dx^2....\end{equation}.

But from time dilation formula in GR, I know this wrong. 

Precisely, according to what I have unsertood, **proper time is the time elapsed in the rest frame of the particle, just like in SR**, so for the interval\begin{equation}
ds^2= g_{00}dt^2 - g_{11}dx^2....\end{equation}, if two events happen at the same place 
\begin{equation}ds^2 = g_{00}dt'=g_{00}d\tau^2\end{equation} ( by definition just as in the SR case). 

Why is this wrong. **Is my reasoning that proper time is the time measured in the rest frame wrong**\
                                     
                                     or\
 **is that the time coordinate in a general metric not represent time measured by any clock**.