= Solution
We prove the <Weinstein neighborhood theorem> with the canonical sign convention $\omega_0=-d\alpha$. The essential first step is to match the two <symplectic forms> as bilinear forms on the entire tangent space along $X$, not only after pulling them back to $TX$.
Here is the needed <symplectic splitting along a Lagrangian submanifold>. Write $E=TM|_X$ and $L=TX$. Choose a smooth <vector bundle> complement $N$ of $L$, for example using a <Riemannian metric>. Since $L$ is a <Lagrangian subspace> in every fiber, the pairing map
$$
N\longrightarrow L^*,\qquad v\longmapsto\bigl(u\longmapsto\omega(u,v)\bigr)
$$
is an isomorphism: its kernel is $N\cap L^\omega=N\cap L=0$, and both bundles have rank $n$. Let $s:L^*\to N$ be its inverse and put $b(\eta,\zeta)=\omega(s\eta,s\zeta)$. Define $a:L^*\to L$ by
$$
\zeta(a\eta)=-\tfrac12 b(\eta,\zeta),
$$
and set $\widetilde s=s+a$. Nondegeneracy of the dual pairing gives a unique smooth $a$. Since $b$ is alternating,
$$
\omega(\widetilde s\eta,\widetilde s\zeta)
=b(\eta,\zeta)-\eta(a\zeta)+\zeta(a\eta)=0.
$$
Thus $K=\widetilde s(L^*)$ is a smooth <Lagrangian complement> to $L$. The map
$$
A:TX\oplus T^*X\longrightarrow TM|_X,\qquad
A(u,\eta)=u+\widetilde s\eta
$$
is a <vector bundle isomorphism>, is the identity on $TX$, and satisfies
$$
\omega(A(u,\eta),A(v,\zeta))=\zeta(u)-\eta(v)
=\omega_0((u,\eta),(v,\zeta)).
$$
The last equality uses the canonical splitting of the tangent bundle of $T^*X$ along its <zero section>.
Choose a <Riemannian metric> near $X$ for which $K$ is orthogonal to $TX$. The <tubular neighborhood> construction using its <Riemannian exponential map> gives a <diffeomorphism>
$$
\psi:W\subset T^*X\longrightarrow W'\subset M,\qquad
\psi(q,p)=\exp_q(\widetilde s_qp),
$$
after shrinking around the <zero section>. It restricts to the given inclusion of $X$ and has differential $A$ along it. Consequently $\beta=\psi^*\omega$ and $\omega_0$ agree pointwise as ambient bilinear forms at every point of the <zero section>.
We use the following precise <relative Moser theorem>. If two closed <symplectic forms> $\gamma_0,\gamma_1$ near a compact embedded submanifold $S$ agree as ambient bilinear forms along $S$, then, on smaller neighborhoods, there is a <diffeomorphism> $\kappa$ fixing $S$ pointwise with $\kappa^*\gamma_1=\gamma_0$. One sufficient version allows a symplectic path $\gamma_t$ with $\partial_t\gamma_t=d\eta_t$ and $\eta_t=0$ as ambient covectors along $S$: solve $\iota_{Z_t}\gamma_t=-\eta_t$ and use its flow. The <vector field> vanishes on $S$, so the flow fixes $S$, and compactness permits a common smaller neighborhood for the full time interval.
For completeness, all hypotheses can be checked directly here. Put $\delta=\beta-\omega_0$ and $\gamma_t=\omega_0+t\delta$. At the <zero section>, every $\gamma_t$ equals $\omega_0$. The <nondegenerate> property is open, and compactness of $X\times[0,1]$ allows a single smaller neighborhood on which all $\gamma_t$ are <symplectic forms>. Choose it fiberwise star-shaped. Write $H_s(q,p)=(q,sp)$ and let $E$ be the vertical radial <vector field>. The <radial homotopy primitive near a zero section>
$$
\eta=\int_0^1 s^{-1}H_s^*(\iota_E\delta)\,ds
$$
is smooth, is zero as an ambient covector along the <zero section>, and satisfies
$$
d\eta=\delta-H_0^*\delta=\delta.
$$
The <radial homotopy operator> gives this identity, using $d\delta=0$ and $H_0^*\delta=\pi^*i_0^*\delta=0$. Smoothness at $s=0$ follows because contraction with $E$ supplies a factor of $s$ after evaluation at $(q,sp)$; the vanishing of $\delta$ along $X$ supplies additional vanishing. Therefore solve $\iota_{Z_t}\gamma_t=-\eta$. By <Cartan's magic formula>, its <flow maps> $\kappa_t$ satisfy
$$
\frac d{dt}(\kappa_t^*\gamma_t)
=\kappa_t^*\bigl(\delta+d(\iota_{Z_t}\gamma_t)\bigr)=0.
$$
After shrinking the starting neighborhood, these flows exist for $0\leq t\leq1$ and fix $X$. This is a relative local construction, not an assertion of completeness throughout the noncompact <cotangent bundle>.
Finally set $\varphi=\psi\circ\kappa_1$, let $U_0$ be its sufficiently small domain, and let $U=\varphi(U_0)$. Then
$$
\boxed{\varphi^*\omega=\kappa_1^*\beta=\omega_0,
\qquad \varphi\circ i_0=i.}
$$
Thus the map is a <symplectomorphism> of neighborhoods and agrees with the specified inclusion at every point of $X$.
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