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.