Omission of locality axiom in some definitions of sheafs

I'm currently reading about sheaf theory and am confused about possible conflicting definitions of a sheaf. The common one I've come across in courses on Algebraic Geometry and also on Wikipedia is the following: Let $$\mathcal{F}$$ be a presheaf of sets on a topological space $$X$$, $$\mathcal{F}$$ is a sheaf if it fulfills two additional conditions:

1. Locality: Let $$U = \bigcup_i U_i$$ be an open cover, $$s, t \in \mathcal{F}(U)$$ be sections such that $$\rho_{U_i}(s) = \rho_{U_i}(t) \forall i$$,then $$s=t$$
2. Gluing: Let $$U = \bigcup_i U_i$$, $$s_i \in \mathcal{F}(U_i)$$ be sections such that $$\rho_{U_i\cap U_j}(s_i) = \rho_{U_i\cap U_j}(s_j) \forall i,j$$, then $$\exists s \in \mathcal{F}(U)$$ with $$\rho_{U_i}(s) = s_i \forall i$$

However on the stacks project, only the gluing condition is mentioned.

I was wondering if there was a reason for this discrepancy and would appreciate any help!

The stacks project requires the glued section $$s$$ to be unique! This uniqueness corresponds to the locality condition, because both $$s$$ and $$t$$ are gluings of the local sections $$\rho_{U_i}(s) = \rho_{U_i}(t)$$, so the uniqueness implies $$s = t$$.