2. The most important thing that a property named "locally P" should obey is that it be local rather than global. In the case of topological spaces, this means that if a topological space X is locally P at some point $x_0$, and we have another topological space Y which agrees with X at a neighbourhood of $x_0$ (e.g. Y could be the restriction of X to a neighbourhood of $x_0$, with the relative topology), then Y should be locally P at $x_0$ as well. Both local compactness and local connectedness, as defined traditionally, have this locality property. On the other hand, the property "$x_0$ has at least one connected neighbourhood" is not local (consider for instance $({\bf Q} \times {\bf R}) \cup \{\infty\}$ in the Riemann sphere ${\bf R}^2 \cup \{\infty\}$, which is a globally connected space which is locally identical near the origin to ${\bf Q} \times {\bf R}$ in ${\bf R}^2$, which has no connected open sets), and thus this property does not deserve the name of "local connectedness at $x_0$".
2. The most important thing that a property named "locally P" should obey is that it be local rather than global. In the case of topological spaces, this means that if a topological space X is locally P at some point $x_0$, and we have another topological space Y which agrees with X at a neighbourhood of $x_0$ (e.g. Y could be the restriction of X to a neighbourhood of $x_0$, with the relative topology), then Y should be locally P at $x_0$ as well. Both local compactness and local connectedness, as defined traditionally, have this locality property. On the other hand, the property "$x_0$ has at least one connected neighbourhood" is not local, and thus this property does not deserve the name of "local connectedness at $x_0$".