Example
If we use the Stacks project definition of a coherent module, \(\mathcal O_X\) is not necessarily itself a coherent \(\mathcal O_X\)-module. This is quite different from the situation in Hartshorne.
Let \(X = \Spec R\) where \(R = k[x,y,z_i,w_i]_{i\in \mathbb N}/\langle xz_i - yw_i\rangle\). We claim that \(\mathcal O_X = R\) is not a coherent \(R\)-module. To see this, choose \(x,y\in R\) and consider the map \(\varphi:R^2 \to R\) defined \((1,0) \mapsto x\) and \((0,1)\mapsto y\). The kernel of this map is then infinitely generated by elements of the form \((z_i, -w_i)\) in \(R^2\), hence \(\ker\varphi\) is not a finite type \(R\)-module and \(R\) is not coherent.
To get an equivalence of Stack’s and Hartshorne’s definitions of coherent, you need to assume \(X\) is locally Noetherian.