What the presentist affirms, roughly, is that only (temporally) present items exist: there are no nonpresent existents. The anti-presentist denies this, maintaining that there are nonpresent existents. Now there is no genuine dispute here unless the identity of the presentist thesis is perfectly clear and the anti-present is denying that very thesis.
Following some earlier suggestions of Peter Lupu, I will now try to formulate this dispute using the concept nonpresent existent. I will use 'NPE' to refer to this concept, a concept we may assume both the presentist and his opponents understand. A nonpresent existent, by stipulative definition, is one that exists in time, but is either merely past or merely future. Using NPE, presentism and anti-presentism may be defined as follows:
P. NPE is not instantiated
AP. NPE is instantiated.
The dispute, then, is about whether NPE is instantiated. NPE is a concept both parties understand. So it is common ground. The dispute is not about this concept, but about whether it is instantiated.
But note that 'is' occurs in both formulations. Does it have exactly the same sense in both (P) and (AP)? If not, then the common ground afforded us by NPE avails us nothing.
Yesterday (see link below) I distinguished five time-related senses of 'is.' Starting with (P), which sense of 'is' is operative in it? We can right away exclude the 'is' of atemporality since presentism is a thesis about temporal items. We can also exclude the 'is' of temporal presentness. The presentist cannot be charitably construed as saying that NPE is not now instantiated, for that is trivially true.
The 'is' of omnitemporality is not a suitable candidate either. For if NPE is not instantiated at every time, then we have quantification over times. But one cannot quantify over what does not exist. So nonpresent times exist. But if so, then NPE is instantiated, contrary to what the presentist intends.
On the disjunctively detensed reading of 'is', the presentist is saying that NPE was not instantiated or is not instantiated or will be not instantiated, and the anti-presentist is saying that the NPE was instantiated or is instantiated or will be instantiated.
Does this do the trick? At the moment I cannot see that it doesn't. But time is the hardest of nuts to crack and my 'nutcracker' may not be up to the job . . . .