Ehrenfeucht-Fraïssé games (Ajtai-Fagin in fact) for regular languages.
We will give a winning strategy for Delilah. Let Samson choose his $c$ and $m$. Then Delilah chooses $w = a^n b^n$ for a large $n$ to be determined later. Let Samson choose his subsets $C^w_1,\ldots,C^w_c$ which we view as a coloring of the positions of $w$ with $2^c$ colors. Let $w'$ denote this colored word. The goal of Delilah at this point is to find a segment $w'[i,\ldots,j]$ of $w'$ with the following properties for an $r$ and a $t$ to be chosen later:


$0 \leq i \leq j \leq n$ (thus the segment belongs to the first part of $w'$),
the $r$-neighborhoods (neighborhoods of radius $r$) of $i$ and $j$ in $w'$ are isomorphic,
for every $k \in [i,\ldots,j]$, the $r$-neighborhood of $k$ in $w'$ appears as $r$-neighborhood of at least $t$ other positions of $w'$.


If she manages to do that, then she will choose her colored word to be
$$
v' = w'[0,\ldots,i-1] w'[i,\ldots,j]^2 w'[j+1,\ldots,2n-1].
$$ 
If $v$ is the ${a,b}$-word underlying $v'$, it will follow that $v$ does not belong to $L$ (because we pumped a non-empty segment of $a$'s), and Delilah has a winning strategy in the $m$-turn EF game on $w'$ and $v'$ (this follows from Hanf's Theorem if $r$ and $t$ are sufficiently large with respect to $c$ and $m$; see Theorem 1.4.1 in Ebinghaus and Flum book "Finite Model Theory").

Thus, it remains to show that if $n$ is sufficiently large with respect to $c$, $m$, $r$ and $t$ we can find a segment $w'[i,\ldots,j]$ as above. But this follows from a standard pigeonhole argument using the fact that the number of ismorphism types of $r$-neighborhoods is finite.

This worked for successor structures. With a linear order it will be a bit harder but I didn't think much about it.

Note that, not surprisingly, this argument looks a bit like "pumping" argument in automata. However, it is not as silly as just translating the formula to an automaton. I think it counts as a model-theoretic argument.