An observer in general relativity is defined as a future directed timelike worldline \begin{align*} \gamma:I \subset \mathbb R &\to M \\ \lambda &\mapsto \gamma(\lambda) \end{align*} together with an orthonormal basis $e_a(\lambda) \in T_{\gamma(\lambda)}M$ where $e_0(\lambda)= v_{\gamma, \gamma(\lambda)}$ and \begin{align} g(\gamma(\lambda))(e_a(\lambda),e_b(\lambda))=\eta_{ab}~. \qquad (1) \end{align} Here, $v_{\gamma, \gamma(\lambda)}$ is the velocity of the worldline $\gamma$ at the point $\gamma(\lambda)\in M$ and $g$ is the metric tensor field on $M$. The time measured by the clock carried by this observer between events $\lambda_0, \lambda_1$ is defined as \begin{align} \tau_\gamma = \int_{\lambda_0}^{\lambda_1} d\lambda \sqrt{g(v_{\gamma, \gamma(\lambda)},v_{\gamma, \gamma(\lambda)})}~. \end{align} However, \begin{align} g(v_{\gamma, \gamma(\lambda)},v_{\gamma, \gamma(\lambda)}) = g(e_0(\lambda),e_0(\lambda))=1 \qquad (2) \end{align} which follows from the requirement of eq.(1). We are using signature $(+,-,-,-)$.
This is all standard definition. Suppose, we have another observer $\delta$: \begin{align*} \delta:I \subset \mathbb R &\to M \\ \lambda &\mapsto \delta(\lambda) \end{align*} and the time measured by his clock between the same two events $\lambda_0, \lambda_1$ is \begin{align} \tau_\delta = \int_{\lambda_0}^{\lambda_1} d\lambda \sqrt{g(v_{\delta, \delta(\lambda)},v_{\delta, \delta(\lambda)})}~. \end{align} From equations (1) and (2), we get $\tau_\gamma = \tau_\delta$ and this will be true for all observers measuring time between $\lambda_0, \lambda_1$.
However, I know that my conclusion is wrong. Can you point out where I went astray?
Edit: I am trying to make the situation I am referring to clearer.
$\gamma(\lambda)$ and $\delta(\lambda)$ meeting at points $p,q \in M$">
(Sorry for this huge picture. I wanted to make it smaller, but couldn't figure out how to go about it.) This picture shows the two observers $\gamma$ and $\delta$ defined above. Both worldlines are parameterised by the same parameter $\lambda$. This need not be the case but I choose it to convey my point. I wish to determine the proper time measured by observers $\gamma$ and $\delta$ between events $p$ and $q$ in the spacetime manifold $M$. \begin{align} p =& \ \gamma(\lambda_1) = \delta(\lambda_1) \\ q =& \ \gamma(\lambda_2) = \delta(\lambda2) \end{align} This scenario is possible, isn't it? I do not see why $\tau_\gamma$ and $\tau_\delta$ need to be the same. (In fact, in the twin paradox, for example, we see this explicitly.) However, from equation (1) and the deduction above, it follows that $\tau_\gamma = \tau_\delta$. This is my confusion.
Note From the definition of an observer in general relativity, the observer worldline seems to be always parameterised by its propertime. But the propertime measured by two observers between the same two events need not be the same, isn't it?