(1) Yes, your recollection is correct, and
(2) It’s unclear if Arpaio has accepted his pardon in a legally meaningful way yet.
No, no, you’re generally on the money here: Arpaio IS trying to have it both ways.
Now, there are some subtleties in what he’s arguing, to be sure, that distinguish his request from earlier cases.
It might be helpful, to understand what he’s arguing, to consider this possibly alarming fact: if you’re convicted of a crime, but die during the pendency of an appeal, the conviction is vacated.
Why? Because your death basically moots the appeal, and since the appeals process is part of the guarantees of fairness that our system provides, it would be unfair to saddle the dead man’s records with a possibly flawed conviction.
So, as far as I can tell, Arpaio is trying to apply that logic to his present circumstance. He says, in effect, that he intended to appeal his conviction, and the appeal would have vindicated him, but now that the pardon exists, there’s no point in any appeal, because the pardon makes it moot, so the conviction itself should be erased from the memories of humankind.
But in my view, this motion is doing what you said: trying to have it both ways. Arpaio seeks the foreclosing effect on prosecution a pardon has, but also wants to avoid the admission of guilt that comes with accepting it.