Let $P$ be a class of topological spaces (for example, compact spaces).
The class of locally $P$ spaces consists of those spaces in which every point has a neighborhood basis consisting of $P$ spaces. The class of weakly locally $P$ spaces consists of those spaces in which every point has a neighborhood in $P$.
Question. What are some categorical properties of the category of locally $P$ spaces which are not shared by the category of weakly locally $P$ spaces, or vice versa? If necessary, assume that $P$ is closed under suitable operations.
This could shed some light on the question which of the two definitions of locally $P$ spaces is more "natural". (And you already might guess my preference.)
Let's start with the obvious: if a topological space is $P$ then it is weakly locally $P$ but not necessarily locally $P$. For example, every connected space is weakly locally connected but not necessarily locally connected. On the other hand, more or less by definition, a topological space is locally $P$ if and only if every open subspace of it is weakly locally $P$. Thus, locally $P$ coincides with weakly locally $P$ if and only if every open subspace of every $P$ space is weakly locally $P$. For example, a topological space is locally $T_1$ (resp. Hausdorff) if and only if it is weakly locally $T_1$ (resp. Hausdorff). The best situation is when weakly locally $P$ and locally $P$ coincide.
It is easy to see that finitary products of weakly locally $P$ spaces are weakly locally $P$ if and only if finitary products of $P$ spaces are weakly locally $P$. Moreover, if finitary products of weakly locally $P$ spaces are weakly locally $P$, then finitary products of locally $P$ spaces are locally $P$. The situation with pullbacks seems more complicated. In the case where open subspaces of $P$ spaces are $P$, one can use the usual patching argument to show that the category of locally $P$ spaces (= weakly locally $P$ spaces here) is closed under pullbacks in $\mathbf{Top}$ if the category of $P$ spaces is. A more careful version of this argument allows us to drop the hypothesis that open subspaces of $P$ spaces are $P$, but then we can only conclude for locally $P$ spaces. It is not obvious to me whether or not there is a way to make the argument work for weakly locally $P$ spaces.