This is a two-dimensional intension, since there are two distinct roles that possibilities play here: as a context of use, and as a circumstance of evaluation (a possible situation relative to which we evaluate whether the relevant object exists or the relevant kind or property is instantiated).
Since with respect to that circumstance of evaluation the terms will differ in reference, their intensions which are just functions from circumstances of evaluations to referents must also differ.
But I left the notion of a circumstance of evaluation rather underspecified.
A single proposition may be true with respect to one circumstance of evaluation and false with respect another.
On the view being discussed, an expression like "according to the Greek myth" is, qua intensional operator, a circumstance-shifting operator, one that shifts the circumstance of evaluation of the sentence following it.
However, both are necessary truths: like any truths of mathematics, neither depends on special features of the actual world, but rather both are true with respect to every circumstance of evaluation.
So we know that expressions are associated with characters, which are functions from contexts to contents; and we know that contents are things which, for each circumstance of evaluation, determine a reference.
(Note that "old-style contextualism" can also be stated in Kaplan's framework; it involves variation in content with respect to the context of utterance rather than in truth value with respect to the circumstance of evaluation).
Borrowing formalism introduced by (Davies and Humberstone 1980), we can say that a sentence is always evaluated with respect to an ordered pair of possible worlds, <wx, wy>, where the first member of the pair represents the world designated as actual and the second represents a circumstance of evaluation.
But according to the 2D empiricist, you can test whether your implicit understanding of 'water' is structured in the relevant way by considering possible situations in two different roles: as your actual environment or as a mere counterfactual circumstance of evaluation (Jackson 1998a, ch. 2).
For a critical survey of other proposed tests for context-sensitivity, see Cappelen & LePore (2005), Part I. Above, in §2.1.5, I introduced the idea of an expression determining a reference, relative to a context, with respect to a particular circumstance of evaluation.
