= Solution
An <affine connection> on the <tangent bundle> of a <smooth manifold> assigns a <vector field> $\nabla_XY$ to two <vector fields>, is $\mathbb R$-bilinear, and satisfies
$$
\nabla_{fX}Y=f\nabla_XY,\qquad
\nabla_X(fY)=X(f)Y+f\nabla_XY.
$$
Thus its first slot is <tensorial> and its second obeys the <Leibniz rule>. Its action on <covectors> is defined by differentiating the pairing, and this extends it to <tensor fields>. A <torsion-free connection> has $\nabla_XY-\nabla_YX=[X,Y]$.
Assume that $g$ is a smooth <nondegenerate> <metric tensor>. <Metric compatibility> and a vanishing <torsion tensor> imply the <Koszul formula>,
$$
2g(\nabla_XY,Z)=Xg(Y,Z)+Yg(Z,X)-Zg(X,Y)
-g(X,[Y,Z])+g(Y,[Z,X])+g(Z,[X,Y]).
$$
The right-hand side is determined by $g$ and the <Lie bracket of vector fields>. Nondegeneracy determines $\nabla_XY$ uniquely; conversely this formula defines an <affine connection> with both required properties, proving existence as well as uniqueness. In a <coordinate frame> the brackets vanish. Combining the three differentiated <metric tensor> identities gives
$$
\boxed{\Gamma^a{}_{bc}
=\frac12g^{ad}(\partial_bg_{dc}+\partial_cg_{db}-\partial_dg_{bc}).}
$$
This is the <Levi-Civita connection>. Positive definiteness is unnecessary: the argument also works for a <pseudo-Riemannian metric>.
For the other <affine connection> define $S(X,Y)=\bar\nabla_XY-\nabla_XY$. The <derivative> of $f$ cancels between the two <Leibniz rules>, so $S$ is $C^\infty$-linear in both slots. The <difference of affine connections is a tensor>: here $S$ is a section of $TM\otimes T^*M\otimes T^*M$ with components $\bar\Gamma^a{}_{bc}-\Gamma^a{}_{bc}$. Both <affine connections> are <torsion-free>, so $S(X,Y)=S(Y,X)$.
Agreement of <geodesics> means agreement of their unparametrized curves; different <affine parameters> are allowed. For every nonzero tangent vector $v$, existence of a <geodesic> with that initial velocity implies that $S(v,v)$ is parallel to $v$. The <symmetric bilinear diagonal-parallel lemma> now determines $S$. Choose a <basis> $e_i$ and write $S(e_i,e_i)=\alpha_i e_i$. Applying the parallelism condition to $e_i+e_j$ and $e_i-e_j$ gives
$$
S(e_i,e_j)=\frac12\alpha_j e_i+\frac12\alpha_i e_j.
$$
With $V(e_i)=\alpha_i/2$, <bilinearity> yields
$$
\boxed{S^a{}_{bc}=\delta^a_bV_c+\delta^a_cV_b
=2\delta^a{}_{(b}V_{c)},\qquad
V_c=\frac{S^a{}_{ac}}{n+1},\quad n=\dim M.}
$$
This is <projective equivalence of affine connections>. It also covers dimension one. For two <Levi-Civita connections>, the <projective covector from metric volume densities> gives the explicit answer
$$
\boxed{V_c=\frac{1}{2(n+1)}
\partial_c\log\left|\frac{\det\bar g}{\det g}\right|.}
$$
Indeed $\Gamma^a{}_{ac}=\partial_c\log\sqrt{|\det g|}$. The <determinant> ratio is a <scalar>, so this expression is a genuine <covector>, although either <determinant> alone is coordinate-dependent. This <determinant> expression requires nondegeneracy of $\bar g$; the difference-tensor and earlier <trace> formula require only the two torsion-free <affine connections>.
Conversely, the displayed <tensor> gives $\bar\nabla_{\dot\gamma}\dot\gamma=2V(\dot\gamma)\dot\gamma$ along an affinely parametrized $\nabla$-<geodesic>. Choosing a new parameter $t(\tau)$ satisfying $t''/t'=2V(\dot\gamma)$ removes that tangential acceleration. This <geodesic reparametrization under projective equivalence> proves sufficiency. If agreement were required with the very same <affine parameter>, polarization would instead force $S=0$ and $V=0$.
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