What is the correct definition of hydration enthalpy and why is it always negative?
Hydration is defined as the following process (with any salt but using copper(II) sulphate as an example):

$$\ce{CuSO4 + 5 H2O -&gt; CuSO4 . 5 H2O (s)}\tag{1}$$

This reaction will only happen if the associated enthalpy $\Delta_\mathrm{hyd} H^0$ — or more precisely, the associated Gibbs free energy $\Delta_\mathrm{hyd} G^0$ is negative. For some compounds, that is not the case; for example sodium chloride does not form hydrates: reaction (2) does not occur spontaneously.

$$\ce{NaCl + n H2O} \nrightarrow \ce{NaCl . n H2O}\tag{2}$$

Thus, we cannot measure a hydration enthalpy. If we can measure it, the process must be spontaneous and thus the enthalpy negative. Thus, all measurable hydration enthalpies are negative.

Sometimes, salts can form multiple hydrates. However, not every hydrate is always possible. Hydration enthalpies only exist for those hydrates which are possible.



Your definitions are basically identical only that they do not measure hydration enthalpies but solvation enthalpies. That is the process as shown in equation (3).

$$\ce{CuSO4 (+ H2O) -&gt; Cu^2+ (aq) + SO4^2- (aq) (+ H2O)}\tag{3}$$

For that process, the enthalpy does not have to be negative. In fact, quite a few ionic compounds dissolve endothermicly, e.g. $\ce{NH4Cl}$.