[A substantive distinction between presentism and eternalism] cannot be formulated in nonindexical terms. That is why I have formulated [presentism] with the aid of the indexical 'presently'.
Others have tried with elaborate means to formulate something which could reasonably be called a non-indexical version of presentism. Matthew Farr, for example, is currently exploring a strategy involving two temporal dimensions. (This note was inspired by a talk he gave at the University of Sydney on 25/3/13 called 'Supertemporal Ontology and the "Triviality" Problem'.)
The issue is obviously interesting, assuming that the question of the meaning of presentism is interesting. One special connection in which it is interesting is this: if there is no non-indexical core of presentism, then perhaps the metaphysical dispute between presentism and non-presentism can be dissolved, on the grounds that presentism is not a distinctive thesis about the structure of reality, but something else (a "view from inside", or something).
I am certainly sympathetic to the idea that the debate in temporal ontology (between presentism, eternalism, the growing block view etc.) is something which should be dissolved or transcended. I seriously doubt that this is a debate about some real subject matter, let alone that it is a debate about some real subject matter where one of the positions is right and the others wrong. I think the confusion here is deep, philosophically important, and deserves to be investigated carefully, not just dismissed. However, I do not think that the idea that the debate cannot be formulated non-indexically is a way to make progress on this, because I think that idea is wrong.
My suggestion here is that we can analyse presentism as the conjunction of two claims, one of which is non-indexical and incompatible with eternalism, growing blockism and shrinking blockism, the other of which is indexical but agreed to by all these parties.
(P) Only the present moment exists.
may be analysed as
(P-conj) Only one moment exists, and this moment exists (where 'this' indicates the present moment).
Call this the conjunction analysis. The non-indexical core of presentism, on this suggestion, is simply the first conjunct: monism about moments (or times, instants, timeslices or whatever).
This non-indexical core is, of course, compatible with strange propositions like:
(S) Only one moment exists, and it is the moment of Napoleon's birth.
But so what? No one believes that, and everyone (presentist, eternalist, growing blockist, shrinking blockist) believes that the present moment exists. Forget the label 'presentism' - look at the conjunction analysis, and it becomes clear that what is really distinctive about this view - i.e. what distinguishes it from other actual contenders - is its monism.
Hinchliff, M. 2000. 'A defense of presentism in a relativistic setting', Philosophy of Science 67, pp. S575-S586.