= Solution
Let $f(x)=(f_1(x),\ldots,f_p(x))^T$ and let an <approximate experimental design> $\xi$ specify the proportions of observations allocated to points of the design region. Its normalized <information matrix of an experimental design> is
$$
M(\xi)=\int f(x)f(x)^T\,d\xi(x).
$$
For $N$ independent equal-variance observations realizing those proportions, $X^TX=NM(\xi)$, and the <least-squares estimator> has <covariance matrix> $\sigma^2M(\xi)^{-1}/N$. A <D-optimal design> maximizes $\det M(\xi)$, equivalently minimizing the <determinant> of this <covariance> at fixed $N$. A <G-optimal design> minimizes the largest <variance of a fitted regression mean> over the design region, equivalently
$$
\sup_x d(x,\xi),\qquad d(x,\xi)=f(x)^TM(\xi)^{-1}f(x),
$$
where $d$ is the <design sensitivity function>. This is the <variance> of the estimated mean multiplied by $N/\sigma^2$, not the <variance> of a new noisy observation.
For continuous regressors spanning a $p$-dimensional parameter space on a compact design region, the <general equivalence theorem for optimal design> states that, among approximate designs with nonsingular information <matrices>, the following are equivalent: being a <D-optimal design>, being a <G-optimal design>, and
$$
\boxed{\sup_x d(x,\xi)=p.}
$$
At every support point of an optimal design, $d=p$. To see the criterion, the sensitivity average is $\int d\,d\xi=\operatorname{tr}(M^{-1}M)=p$, so the supremum is at least $p$. Adding an infinitesimal allocation at $x$ has directional derivative $d(x,\xi)-p$ for $\log\det M$. At a D-optimum each derivative is nonpositive. Conversely, if all sensitivities are at most $p$, the tangent bound for the concave function $\log\det M$ shows that no competing design increases it. A nonsingular D-optimum exists by the spanning assumption and compactness of the information-matrix set, so the smallest possible G-value is also $p$. Equality at support points follows from the average. The theorem refers to the full approximate-design class; constrained integer allocations or mandated augmentations require separate comparison.
For the linear mean $\beta_0+\beta_1x_1+\beta_2x_2$, take $f(x)=(1,x_1,x_2)^T$. Equal replication at the four corners gives
$$
X^TX=4mI_3.
$$
All odd corner sums and the sum of $x_1x_2$ vanish, while each squared regressor sum is $4m$. If $\bar Y_{ab}$ is the mean of the $m$ observations at $(a,b)$, $a,b\in\{-1,1\}$, the <least-squares normal equations> give
$$
\boxed{\widehat\beta=\frac14\sum_{a,b\in\{-1,1\}}\bar Y_{ab}
\begin{pmatrix}1\\a\\b\end{pmatrix},\qquad
\operatorname{Cov}(\widehat\beta)=\frac{\sigma^2}{4m}I_3.}
$$
In particular the <intercept> is the average of the four cell means, and each slope is the average of its sign-weighted cell means. Independence and equal error <variance> suffice for this <covariance> formula; <normal> errors are not needed for it.
The normalized corner design has $M=I_3$, so its sensitivity is
$$
d(x,\xi)=1+x_1^2+x_2^2\le3\qquad\text{on }[-1,1]^2,
$$
with equality at all four corners. Since $p=3$, the <general equivalence theorem for optimal design> proves that \b[the equally weighted corner design is D-optimal and G-optimal].
For the augmentation, all $m$ added observations share the regressor $f=(1,x_1,x_2)^T$. The enlarged <design matrix> satisfies
$$
\widetilde X^T\widetilde X=m(4I_3+ff^T).
$$
By the <matrix determinant lemma>,
$$
\begin{aligned}
\det(\widetilde X^T\widetilde X)
&=m^3\det(4I_3)\left(1+f^T(4I_3)^{-1}f\right)\\
&=64m^3\left(1+\frac{1+x_1^2+x_2^2}{4}\right)
=16m^3(5+x_1^2+x_2^2).
\end{aligned}
$$
For the permitted nine points, the <determinant> is $80m^3$ at the center, $96m^3$ at an edge midpoint and $112m^3$ at a corner. Therefore
$$
\boxed{(x_1,x_2)\in\{(-1,-1),(-1,1),(1,-1),(1,1)\},
\qquad\det(\widetilde X^T\widetilde X)=112m^3.}
$$
All four corners tie for the best mandated single-point augmentation. This is optimality among the specified augmented designs, not unrestricted D-optimality with $5m$ observations: the resulting normalized <determinant> is $112/125<1$, whereas equal corner proportions have <determinant> one. The restriction that the extra $m$ runs all occur at one point is what prevents recovering equal proportions.
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