The Stacks project

Maybe this is trivial to ask but is there a definition for the notion $\mathrm{colim}\; \mathcal{X}_{T_i}$ (i.e. a colimit of groupoids)? My understanding is that this is just a suggestive notation (and the meaning of the equivalence $\mathrm{colim}\;\mathcal{X}_{T_i}\to \mathcal{X}_T$ is explained in the paragraph followed the above definition). Is this correct or I am missing something?

First of all: yes, please see the explanation following the definition for what the definition means. OTOH you can define colimits of directed systems of groupoids exactly as suggested by this explanation and for the purposes of the Stacks project, this is the correct definition. You can also define $\text{colim} \mathcal{X}_i$ if you just assume: (1) $I$ is a directed set (Definition 4.21.1), (2) for each $i \in I$ we have a groupoid $\mathcal{X}_i$, and (3) for each $i \leq j$ we have a functor $F_{ij} : \mathcal{X}_i \to \mathcal{X}_j$ with $F_{ii} = \text{id}_{\mathcal{X}_i}$, and (4) for every $i \leq j \leq k$ a transformation $F_{ik} \to F_{jk} \circ F_{ij}$ such that this data forms a pseudo functor from $I$ to the $2$-category of groupoids, see Definition 4.29.5. The colimit of such a beast constructed more or less in the same way will have a weak universal property which I leave it up to you to clarify.