The standard examples of complete but not model-complete theories seem to be:

- Dense linear orders with endpoints.

- The full theory $\mathrm{Th}(\mathcal{M})$ of $\mathcal{M}$, where $\mathcal{M} = (\mathbb{N}, >)$ is the structure of natural numbers equipped with the relation $>$ (and nothing else, i.e. no addition etc).

Can anyone explain or give a reference to show why any of these two theories are not model-complete, or give another example altogether of a complete but not model complete theory (with explanation)?