I have a book saying,
$\int \delta(x-x')\psi(x)dx = \psi(x')$ where $\psi(x) = \langle x\lvert\psi\rangle$, so our definition of delta function would be $\langle x'\lvert x\rangle = \delta(x-x')$.
However I could find some documents (example; refer to 3. Position Space) saying,
$$\delta(x'-x'') = \langle x'\lvert x''\rangle$$
which corresponds to $\delta(x-x') = \langle x\lvert x'\rangle$.
So the result should be
$$\delta(x-x') = \langle x\lvert x'\rangle (=) \langle x'\lvert x\rangle \tag{1}$$
I think neither of them is an error, because my book uses the definition many times and I have found many documents explaining as $\delta(x-x') = \langle x\lvert x'\rangle$. Is (1) correct?