= Solution
For $d$-dimensional unitaries, $\operatorname{tr}(U^\dagger U)=d$. The relative propagator $W=U^\dagger V$ is unitary, and its eigenvalues lie on the unit circle. The triangle inequality gives $|\operatorname{tr}W|\leq d$, with equality precisely when all eigenvalues have the same <phase>, or $V=e^{i\gamma}U$. Therefore the specified <trace-overlap gate fidelity> lies between zero and one, is insensitive to <global phase>, and equals one exactly when the implemented gate has the same physical action as the target. It compares propagators independently of any particular input state; it is not the identically defined quantity as average state or average gate fidelity.
For target $R_x(\pi)$ and actual $R_x[\pi(1+\epsilon)]$, the relative propagator is $R_x(\pi\epsilon)$. Its trace is $2\cos(\pi\epsilon/2)$, so
$$
\boxed{\mathcal F_1=\left|\cos\frac{\pi\epsilon}{2}\right|=1-\frac{\pi^2}{8}\epsilon^2+O(\epsilon^4),\qquad \alpha=\frac{\pi^2}{8}.}
$$
The absolute value is unnecessary in a sufficiently small neighbourhood of zero error, but belongs in the exact expression.
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