The definition of weakest precondition for a non-deterministic language
First of all, the definition of weakest precondition you've mentioned might be a little bit confusing. Namely, remember that the weakest precondition, designates all states of a program, such that when the given command executes, the given postcondition is satisfied.

Therefore, your initial definition might be a bit confusing, in the sense that the weakest precondition wp(s, Q) is determined really based on a postcondition (which you have as Q) and a command (which you have a c) -- that is all that is needed to define weakest precondition (w.p.).

Having that said, a more natural definition of wp(c, Q) says that:
$$
wp(c, Q) = \{ \sigma \in \Sigma | \text{ex}[c, \sigma]  \models Q \}
$$
the weakest precondition, given c and Q, is a set of all states, such that executing c (denoted simply by ex here) from those states, assuming the execution terminates (we will come back to that later), will lead to some state $\sigma$ for which the postcondition Q is satisfied.

Given that, the weakest precondition you've mentioned, namely


  "wp (c1;;c2) Q s = wp c1 (wp c2 Q) s" 


is a valid formulation for w.p. for compositions, as, naturally, executing c1 you want to end up in a state belonging to the weakest precondition of wp(c2, q), which, in turn, means that after further executing c2 (in the sequence) you will have to satisfy postcondition (again, with the assumption that both terminate).

Now, in terms of this weakest precondition formulation, while the quote


  ensuring that execution of S terminates in a final state satisfying R


might confuse you, this does not say that termination is encoded in the w.p. formulation. Specifically, it is not required: the definition of weakest precondition says, that, if the given command executes and terminates, from those states, the postcondition has to hold.

That is why this is part of something traditionally called "partial correctness" and not "total correctness", as it does not prescribe termination (but rather, effectively, works under assumption of termination).