Broadly speaking, I think your sensibility about extensional formulas being contrasted with intensional features that are represented in possible world semantics is correct. The question opens up a bunch of interesting issues.
It is worth mentioning that there are several different conceptions of possible worlds. Also, there are many different kinds of necessity and we can use modal logics and possible worlds with any of them if we choose. The question of the relationship between sentences and possible worlds depends on what we are trying to do. Are we aiming to express the relationship of logical consequence between formulas in a simple artificial language like first-order classical logic? Or a more complex language that has modal extensions? Or are we aiming for something much more ambitious such as capturing the meaning of modal statements in a natural language?
Starting with the first of these. The standard approach to logical consequence is to divide it into two kinds, syntactic or proof-theoretic, and semantic or model-theoretic. The former is concerned with what we can derive using formal methods, and the latter with what lacks a falsifying interpretation. There is fair amount of literature on how to represent these using modal logic, and while there is no general concensus, John Burgess makes a good case for saying that we can represent the syntactic kind using S4, and the semantic kind using S5. Since your first piece of quoted text uses model-theoretic terminology, this indicates that it is no surprise that there is a parallel between logical entailment and strict implication. Logical entailment in simple cases is correctly represented by S5 strict implication.
Moving on to more complex modal logics. There is expressive value in adding an 'actually' operator to modal logic and distinguishing between what is true from the perspective of the actual world and what is true from the perspective of any world. This allows us to distinguish two kinds of necessity. There is a kind that is expressed by saying that P is necessary at the actual world if it holds in all worlds that are accessible to it, and a kind that is expressed by saying that P holds in every world, no matter which one we designate as actual. Davies and Humberstone maintain that the latter provides a better understanding of logical entailment, particularly if we wish to follow Kripke and others in holding to a theory of names as rigid designators and to allow the existence of a priori contingent propositions and a posteriori necessary propositions.
As an interesting corollary, Brian Weatherson shows how we might use the same distinction to create a unified account of the logic of indicative and subjunctive conditionals. The very short and simplified version is that an indicative conditional is concerned with what holds from the perspective of the actual world, while a subjunctive conditional is concerned with what holds if some other world were the actual world. It is a neat idea, though not without its difficulties.
It is fairly common that talk of possible worlds brings with it some metaphysical assumptions. This widens the distinction between logical entailment and necessity, since metaphysical necessity is quite different in many respects. Moving further in this direction takes us into the realm of two-dimensional semantics. There is a great deal of recent literature on this. Lloyd Humberstone provides a survey of some of the approaches to its logic.
John Burgess, "Which Modal Logic is the Right One?", Notre Dame Journal of Formal Logic, Vol. 40, (1999), pp. 81-93.
Martin Davies and Lloyd Humberstone, "Two Notions of Necessity" Philosophical Studies, Vol. 38, (1980), pp. 1-30.
Brian Weatherson, "Indicative and Subjunctive Conditionals", Philosophical Quarterly, Vol. 51, (2001), pp. 200-216.
Lloyd Humberstone. "Two-Dimensional Adventures", Philosophical Studies, Vol. 118, (2004), pp. 17-65.
