I think the problem lies in the question "what is the Old Republic?" Because, correct me if I am wrong but, continuity-wise, there is only the Old Republic, Galactic Empire (movie era) and New Republic (post-movie, Vong era).
In a sense, the demise of Bane gave way to a "new Republic" but that term cannot be used because it is already in use. I think, eventually, we will see constructs such as 1st Republic, 2nd Republic, etc.
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It was just a reforming of the Old Republic. the New Republic was an entirely different structure, as was the Empire, Alliance of Free Planets, and Galactic Alliance of Free Planets.
When the Old Republic was reformed it was still the same governement with a Chancellor and everything. Thus it's still the Old Republic.
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And yet you claim that it was a different government. I say if it walks like a duck, quacks like a duck, and fly's like a duck, it is a duck. same with the republic.