$\DeclareMathOperator{\Spec}{Spec}$ Let $X$ be an algebraic stack. Is there is a well-defined notion of the residue field of a point $x \in |X|$?

Attempts:

1. Recall that a point on a stack is an equivalence class of morphisms $\Spec k \to X$ from fields $k$. The issue is that it is not clear that there is a minimal choice of $k$ which contains every such morphism to warrant being called the residue field.
2. There is also the notion of a residual gerbe on a stack; but again it is not clear whether this comes with some kind of canonical field of definition which is compatible with 1.
3. If $X$ has a coarse moduli space $X^c$, then one could define the residue field of $x$ to be the residue field of the image of $x$ in $X^c$. This is well-defined, but seems to lose some of the subtle properties of stacks and again it's not clear whether it is compatible with 1. and 2.

I'm happy to assume my stack is sufficiently nice (e.g. smooth, DM,..)

$\DeclareMathOperator{\Spec}{Spec}$ Let $X$ be an algebraic stack. 

Is there is a well-defined notion of the residue field of a point $x \in |X|$?

Attempts:

1. Recall that a point on a stack is an equivalence class of morphisms $\Spec k \to X$ from fields $k$. The issue is that it is not clear that there is a minimal choice of $k$ to warrant being called the residue field.
2. There is also the notion of a residual gerbe on a stack; but again it is not clear whether this comes with some kind of canonical field of definition which is compatible with 1.
3. If $X$ has a coarse moduli space $X^c$, then one could define the residue field of $x$ to be the residue field of the image of $x$ in $X^c$. This is well-defined, but seems to lose some of the subtle properties of stacks and again it's not clear whether it is compatible with 1. and 2.

I'm happy to assume my stack is sufficiently nice (e.g. smooth, DM,..)