Defeinition of robust counterpart in robust optimization

I’m working on robust optimization problems and trying to summarize a bit about some basic ideas.

However, one really confusing thing is the definition of a robust counterpart.

Definition 1.2.5 from Ben-tal’s book, says
begin{equation}
min _{x}left{widehat{c}(x)=sup _{(c, d, A, b) in mathcal{U}}left(c^{T} x+dright): A x leq b forall(c, d, A, b) in mathcal{U}right}
end{equation}

is the robust counterpart of the uncertain linear optimization
begin{equation}
left{min _{x}left{c^{T} x+d: A x leq bright}right}_{(c, d, A, b) in mathcal{U}}.
end{equation}

This sounds like a robust counterpart is the corresponding optimization problem finding the worst-case optimal solution.

However, in his paper “Robust Solutions of Uncertain Linear Programs”, the robust counterpart is defined as reformulating a deterministic nominal optimization problem to an optimization problem taking uncertainty set into account.

Can someone explain a bit how to define robust counterpart?

And also an example is given that
begin{equation}
min x_{1}+x_{2} text { s.t. }left{begin{array}{rr}
frac{1}{2} x_{1}+x_{2} geq 1 \
x_{1}+frac{1}{2} x_{2} geq 1 \
x_{1}+x_{2}=1 \
x_{1}, quad x_{2} geq 0
end{array}right.
end{equation}

is a robust counterpart for
begin{equation}
min x_{1}+x_{2} text { s.t. }left{begin{array}{rr}
a_{11} x_{1}+x_{2} geq 1 \
x_{1}+a_{22} x_{2} geq 1 \
x_{1}+quad x_{2}=1 \
x_{1}, quad x_{2} geq 0
end{array}right.
end{equation}

with $mathcal{U}=left{a_{11}+a_{22}=2, frac{1}{2} leq a_{11} leq frac{3}{2}right}$.
I’m not able to understand why optimization with the selection of $a_{11}$ and $a_{22}$ as $frac{1}{2}$ is a robust counterpart. This selection doesn’t even satisfy the uncertainty constraint set $mathcal{U}$.