Contrary to what is claimed in the comments, I would argue that the definition given in nLab's Idea section is rigorous enough to be an actual definition in a research-level paper, possibly with an additional phrase thrown in like “The sets of objects and morphisms are equipped with the obvious smooth structures that turn this groupoid into a Lie groupoid.”

Let's see how these smooth structures are constructed. Recall that the set of objects is $\{π^{−1}(x)\mid x∈X\}$, i.e., the set of fibers of $P$. Fibers are in a bijective correspondence with points in the base $X$, and the latter is a smooth manifold.

The set of morphisms is $\{f\colon π^{−1}(x)→π^{−1}(y)\mid \text{$f$ is a $G$-equivariant morphism}\}$. A morphism between two $G$-torsors $X→Y$ is uniquely determined by its value $y∈Y$ at some point $x∈X$. That is, for any pair $(x,y)∈X⨯Y$ there is exactly one morphism that sends $x↦y$. The pair $(gx,gy)$ gives rise to the same morphism $X→Y$ as $(x,y)$. It is also easy to see that the converse is true: $(x,y)$ and $(x',y')$ yield the same morphism if there is $g∈G$ such that $(x',y')=(gx,gy)$. Thus, the set of morphisms $X→Y$ is $(X⨯Y)/G$, where $G$ acts on $X⨯Y$ via $g(x,y)=(gx,gy)$. The action of $G$ on $X⨯Y$ is a smooth free proper action, so the quotient $(X⨯Y)/G$ is a smooth manifold and the quotient map $X⨯Y→(X⨯Y)/G$ is a submersion.

From here, we see that the set of all morphisms is $(P⨯P)/G$ and therefore possesses a canonical smooth structure. The source and target maps are surjective submersions by the 2-out-of-3 property.

Contrary to what is claimed in the comments, I would argue that the definition given in nLab's Idea section is rigorous enough to be an actual definition in a research-level paper, possibly with an additional phrase thrown in like “The sets of objects and morphisms are equipped with the obvious smooth structures that turn this groupoid into a Lie groupoid.”

Let's see how these smooth structures are constructed. Recall that the set of objects is $\{π^{−1}(x)\mid x∈X\}$, i.e., the set of fibers of $P$. Fibers are in a bijective correspondence with points in the base $X$, and the latter is a smooth manifold.

The set of morphisms is $\{f\colon π^{−1}(x)→π^{−1}(y)\mid \text{$f$ is a $G$-equivariant morphism}\}$. A morphism between two $G$-torsors $U→V$ is uniquely determined by its value $v∈V$ at some point $u∈U$. That is, for any pair $(u,v)∈U⨯V$ there is exactly one morphism that sends $u↦v$. The pair $(gu,gv)$ gives rise to the same morphism $U→V$ as $(u,v)$. It is also easy to see that the converse is true: $(u,v)$ and $(u',v')$ yield the same morphism if there is $g∈G$ such that $(u',v')=(gu,gv)$. Thus, the set of morphisms $U→V$ is $(U⨯V)/G$, where $G$ acts on $U⨯V$ via $g(u,v)=(gu,gv)$. The action of $G$ on $U⨯V$ is a smooth free proper action, so the quotient $(U⨯V)/G$ is a smooth manifold and the quotient map $U⨯V→(U⨯V)/G$ is a submersion.

From here, we see that the set of all morphisms is $(P⨯P)/G$ and therefore possesses a canonical smooth structure. The source and target maps are surjective submersions by the 2-out-of-3 property.