I mean I think his audience at the time would have, particularly given what was left in, understood it to mean that it wasn't "unnecessary squeamishness"; under the law of the time it was literally unprintable and he'd have faced serious criminal consequences if he'd tried.
Like, look how much trouble they had with publishing Ulysses, decades later: https://en.wikipedia.org/wiki/Ulysses_(novel)#Publication_hi... . And Ulysses had the benefit of being far less obscene than the diaries, and was Great Literature (TM), rather than, frankly, some pervert's private diaries which had never been intended for publication in the first place. I doubt the diaries could have been safely published in full until after the whole Lady Chatterley's Lover thing, at least not in the UK or US (France was more laid back about this sort of thing).
I still think you're doing some special pleading here. Yes, he might have meant people to infer that, but compare with Bowdler at the earlier end of the same century quite simply saying that he'd omitted passages for "propriety."
In Bowdler's case, though, he was doing _unnecessary_ censorship; you could get basically uncensored versions of Shakespeare's work at the time.
Wheatley _could_ maybe have been more explicit and said "I've omitted the illegally obscene bits", but honestly given the legal environment of the time this would have been asking for trouble.
He could have said that he had omitted some passages for impropriety. If Bowdler could say then then so could he - by the standards of his day or of ours. Saying nothing would have been an option too. His actual statement still seems at worst a lie and at best intentionally obfuscated.
I think we'll have to agree to disagree (although it would be very interesting if there was contemporary evidence of his actual thoughts and motivations).