= Solution
Put $p=\gamma(0)$, $v=\dot\gamma(0)$, $a=J(0)$ and $b=D_tJ(0)$. Choose the short curve $c(s)=\exp_p(sa)$ and let $P_s:T_pM\to T_{c(s)}M$ be <parallel transport> along $c$. The tangent vectors
$$
V(s)=P_s(v+sb)
$$
satisfy $V(0)=v$ and $\nabla_sV(0)=b$. Define a <geodesic variation> by
$$
\boxed{F(s,t)=\exp_{c(s)}\bigl(tV(s)\bigr).}
$$
For each fixed $s$ this is an affinely parametrized <geodesic> with initial point $c(s)$ and initial velocity $V(s)$. Although the <Riemannian manifold> need not be complete, smooth dependence for the <geodesic equation> on a neighborhood of the compact reference segment provides a single $\varepsilon>0$ on which $F$ exists for all $|s|<\varepsilon$ and $0\leq t\leq1$. Thus completeness is unnecessary.
Set $X(t)=\partial_sF(0,t)$. The torsion-free <Levi-Civita connection> gives $\nabla_s\partial_tF=\nabla_t\partial_sF$. Differentiating $\nabla_t\partial_tF=0$ with respect to $s$, and commuting the two <covariant derivatives>, yields
$$
D_t^2X+R(X,T)T=0.
$$
Also $X(0)=c'(0)=a$ and $D_tX(0)=\nabla_sV(0)=b$. Uniqueness for the <Jacobi field> equation now gives \b[$X(t)=J(t)$] on the whole segment. This is <realization of Jacobi fields by geodesic variations>.
The printed formula switches the order of its two variables. The consistent notation here is $F(0,t)=\gamma(t)$ and $J(t)=\partial_sF(0,t)$. A parametrized surface here means a smooth variation map; it cannot be required to be an <immersion>, since the allowed <Jacobi field> $J=0$ would already violate that condition along the reference curve.
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