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In Cavalcanti and Skrzypczyk (2017), section IIA, pag. 3, while introducing the idea of quantum steering, the authors introduce the idea of Local Hidden State model as following (emphasis mine):

A LHS model refers to the situation where a source sends a classical message $\lambda$ to one of the parties, say Alice, and a corresponding quantum state $\rho_\lambda$ to the other party, Bob. Given that Alice decides to apply measurement $x$, the variable $\lambda$ instructs Alice’s measurement device to output the result a with probability $p(a|x, \lambda)$. Additionally it is also considered that the classical message $\lambda$ can be chosen according to a distribution $\mu(\lambda)$. Bob does not have access to the classical variable $\lambda$, so the final assemblage he observes is composed by the elements $$\sigma_{a|x}=\int d\lambda \mu(\lambda)p(a|x,\lambda)\rho_\lambda.$$

I'm struggling to grasp the general picture here. Isn't this equivalent to saying that Alice and Bob share a state of the form $$\sum_\lambda |\lambda\rangle\!\langle\lambda|\otimes \rho_\lambda,$$ for some choice of a computational basis $|\lambda\rangle$ for Alice? In particular, it's the wording that "the source sends a classical message to Alice" that confuses me. What is a "classical message" here? Do we just mean that Alice is not allowed to perform measurements or operations that probe the coherence between different states $|\lambda\rangle$, for some choice of computational basis that was chosen beforehand?

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It is equivalent to saying that Alice and Bob share a state of the form $$\rho^{AB} = \int \mathrm{d}\lambda \mu(\lambda) |\lambda\rangle\langle \lambda| \otimes \rho_\lambda, $$ without any restriction whatsoever on Alice's measurements (you forgot the normalisation, without which it is not a valid quantum state). The expression "classical message" only implies that there's no restriction on how the probability distribution $p(a|x,\lambda)$ depends on $\lambda$. Allowing the different $\lambda$ to be associated to orthogonal quantum states is enough to attain full generality. The formulation with the "classical message" is the most usual one, because it's more convenient for making calculations with a computer.

Perhaps it's more enlightening to consider which assumptions go into a quantum steering scenario: nothing is assumed about Alice's measurements, so the only information about her is that she obtains outcome $a$ for input $x$ with probability $p(a|x)$. On the other hand, it is assumed that everything is known about Bob's measurements, so he can use them to perform quantum state tomography on his part of the share quantum state, and recover the conditional states $\rho_{a|x}$. All the information available can then be encoded in the assemblage $\sigma_{a|x} := p(a|x)\rho_{a|x}$. Now one can ask whether this assemblage allows for a Local Hidden State model, or it certifies that Alice and Bob shared entanglement.

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