Massey products are discussed in Section 1.3 of
Fukaya, Kenji. Morse homotopy,
$A_{\infty}$-category, and Floer
homologies. Proceedings of GARC
Workshop on Geometry and Topology '93
(Seoul, 1993), 1--102,
available here (pdf). The Massey products are obtained by counting gradient flow graphs with four external edges and one (finite-, possibly zero-, length) internal edge. Fukaya sketches a construction of an $A_{\infty}$ category whose objects are Morse functions $f$ and with morphisms from $f$ to $g$ given by the Morse chain complex of $f-g$. In particular the Massey products can then be seen as arising from the $A_{\infty}$ structure in a standard formal way.
There is a relation to Lagrangian Floer theory: a Morse function $f:M\to \mathbb{R}$ corresponds to a Lagrangian submanifold $graph(df)$ of $T^*M$ and intersections between $graph(df)$ and $graph(dg)$ are in obvious bijection with critical points of $f-g$. There are results, pioneered by
Fukaya, Kenji; Oh, Yong-Geun.
Zero-loop open strings in the
cotangent bundle and Morse homotopy.
Asian J. Math. 1 (1997), no. 1,
96--180,
which relate the gradient flow graphs appearing in the Morse $A_{\infty}$ operations to the holomorphic curves appearing in the Lagrangian Floer $A_{\infty}$ operations.
Theorem 2 is true, verbatim. I will give an outline of the proof here, since the details make it kind of long. If you would like a detailed write-up and you don't want to do it yourself, DM or email me.
The actual statement that is true is more general: you do not need the boundary $\partial X$ to be contact or for the boundary $\partial L$ to be Legendrian.
Theorem: (Weinstein Neighborhood With Boundary) Let $(X,\omega)$ be a symplectic manifold with boundary $\partial X$ and let $L \subset X$ be a properly embedded, Lagrangain sub-manifold with boundary $\partial L \subset \partial X$ transverse to $T(\partial X)^\omega$.
Then there exists a neighborhood $U \subset T^*L$ of $L$ (as the zero section), a neighborhood $V \subset X$ of $L$ and a diffeomorphism $f:U \simeq V$ such that $\varphi^*(\omega|_V) = \omega_{\text{std}}|_U$.
Proof: The proof has two steps. First, we construct neighborhoods $U \subset T^*L$ and $V \subset X$ of $L$, and a diffeomorphism $\varphi:U \simeq V$ such that:
\begin{equation}
\varphi|_L = \text{Id} \qquad \varphi^*(\omega|_V)|_L = \omega_{\text{std}}|_L \qquad T(\partial U)^{\omega_{\text{std}}} = T(\partial U)^{\varphi^*\omega}
\end{equation}
Here $T(\partial U)^{\omega_{\text{std}}} \subset T(\partial U)$ is the symplectic perpendicular to $T(\partial U)$ with respect to $\omega_{\text{std}}$ (and similarly for $T(\partial U)^{\varphi^*\omega}$. Second, we apply Lemma 1 (below) and a Moser-type argument to conclude the result.
For the first part, the proof proceeds like this. First you pick a metric on $L$ and use the exponential map in the usual way, to get a diffeomorphism $\varphi:U \simeq V$ with $U \subset T^*L$, $V \subset X$ and $\varphi^*\omega_{\text{std}} = \omega$ along $L$. Then we use Lemma 2 below to modify $\varphi$ in a collar neighborhood of $\partial U$ to satisdy $T(\partial U)^{\omega_{\text{std}}} = T(\partial U)^{\varphi^*\omega}$.
The second part is basically identical to the usual Moser argument.
Lemma 1: (Fiber Integration With Boundary) Let $X$ be a compact manifold with boundary, $\pi:E \to X$ be a rank $k$ vector-bundle with metric and $\pi:U \to X$ be the closed disk bundle of $E$. Let $\kappa \subset T(\partial U)$ be a distribution on $\partial U$ invariant under fiber-wise scaling. Finally, suppose that $\tau \in \Omega^{k+1}(U)$ is a $(k+1)$-form such that:
\begin{equation} \label{eqn:fiber_integration_sigma} d\tau = 0 \qquad \tau|_X = 0 \qquad (\iota^*_{\partial X}\tau)|_\kappa = 0\end{equation}
Then there exists a $k$-form $\sigma \in \Omega^k(U)$ with the following properties.
\begin{equation} \label{eqn:fiber_integration_tau} d\sigma = \tau \qquad \sigma|_X = 0 \qquad (\iota^*_{\partial X}\sigma)|_\kappa = 0 \end{equation}
The proof of Lemma 1 just involves examining the proof of the version of the Poincare Lemma in McDuff-Salamon, and checking that the primitive constructed there satisfies the 3rd property. Note that to apply this lemma, you need to show that the characteristic foliation of $\partial(T^*L) \subset T^*L$ is invariant under fiber-scaling, but this is a quite easy Lemma.
Lemma 2: Let $U$ be a manifold and $L \subset U$ be a closed sub-manifold. Let $\kappa_0,\kappa_1$ be rank $1$ orientable distributions in $TU$ such that $\kappa_i|_L \cap TL = \{0\}$ and $\kappa_0|_L = \kappa_1|_L$.
Then there exists a neighborhood $U' \subset U$ of $L$ and a family of smooth embeddings $\psi:U' s\times I \to U$ with the following four properties.
$$
\psi_t|_{\partial L} = \text{Id} \qquad d(\psi_t)_u = \text{Id} \text{ for }u \in L \qquad \psi_0 = \text{Id} \qquad [\psi_1]_*(\kappa_0) = \kappa_1
$$
Furthermore, we can take $\psi_t$ to be $t$-independent for $t$ near $0$ and $1$.
The proof of Lemma 2 is straight-forward.
Best Answer
Basically, Weinstein's theorem says that you can embed $T^*L$ into $M$ like that: $$ T^*L\cong NL\cong \mathcal{T}_L\subseteq M, $$ where $NL$ is the normal bundle and $\mathcal{T}_L$ is a tubular neighborhood, in such a way that the canonical symplectic form on $T^*L$ is the pull-back of $\omega$. So, the "contact counterpart" of above chain of identifications should read $$ J^1K\cong NK\cong \mathcal{T}_K\subseteq Y $$ where now $J^1K$ is the first-order jet bundle of (smooth) functions on $K$. Notice that all manifolds appearing above are $(2n+1)$-dimensional, and all bundles are over $K$ with $(n+1)$-dimensional fibers. I'm sure this observation can be found in the literature about jet spaces and/or contact manifolds.
My guess is that the above embedding, at least locally, can be always found. Concerning the real question, i.e., whether the canonical contact form on $J^1K$ is the pull-back of $\lambda$, I'm not able to answer, but I feel it is not true!