Part of the issue here, I think, is the degree to which a sapient being can ever truly understand the identity process of another. The final chapter of The Persistence of Memory
sort of suggests that this is going to be a dilemma. Geordi is somewhat confused that Data considers his relationship to the old Data to be problematic, difficult to describe. If it has Data's memories, Data's knowledge and Data's feelings regarding, for example, his friends, then as far as Geordi's concerned it's Data, right? But Data himself isn't so sure, because his vantage point is rather different from Geordi's.
I guess the big question is, will the differences between this Data and the old Data be pronounced enough to convince other characters to see him as a new being, close to the old but distinct, or will they insist on viewing him as Data, restored? Or, put another way, are Geordi etc going to insist that this we've truly pressed the reset button because they're pleased to have their friend back and can only see what's identcal, not what's new? I can only assume, based on that final chapter, that the next two books will explore all this in detail.