Those are good points. Surely #1 and #3 are correct.
I think we agree on the central point: that rags like the Grauniad systematically understate the sizes of urbanizations in China, in a variety of ways. We might disagree on why, and what the effect is.
As for #2, the question of what to call a "city" is sort of more about the cultural practices of translators than about patterns of urban growth. People in China mostly don't call Wuxi a "city"; I think they call it a 市 (the administrative unit), a 地级市 (the larger administrative unit Wuxi Prefecture), a 城市 (the social phenomenon), a 都市, or a 城. How the Guardian chooses to describe Wuxi for their English-speaking readers, mostly outside China, is really the question here.
You seem to be suggesting that writers draw the English-language line between "town" and "city" at a larger population level when they're talking about Chinese cities than when they're talking about cities in, say, England. I think you're correct. I picked three smaller "cities" in Jiangsu from Wikipedia to see how the English-speaking press describes them: Pizhou (population 163k; nytimes: "a village in Jiangsu Province in eastern China"), Yangzhong (pop. 344k; Grauniad: "a county-level city in Jiangsu province", "Yangzhong city"), and Jintan (564k; BBC: "the town of Jintan").
Wuxi has probably more than doubled since 02000, as you say, and Birmingham surely hasn't. However, even in 02000 I don't know if it was a "small city" in any absolute sense (rather than compared to Suzhou and Shanghai). Its population was surely already larger and denser than Birmingham's.
Some Western cities do have that kind of powerhouse growth; a bit of searching turns up Toluca, The Woodlands (arguably still too small to be a "city"), Brasilia, and Las Vegas. Their growth rates are in the same neighborhood as Wuxi's over the last 20 years. Brasilia, for example, has grown from 2.0 million in 02000 to 3.0 million today, 4.3 million in the metro area.
However, I'm not sure I understand your concern about how making these comparisons is "just going to confuse people." What kind of information do you think would be more useful for understanding how big Wuxi is than knowing that its population exceeds that of, for example, Mongolia, Puerto Rico, or any metropolitan area in Germany, and that the population density in its center is half that of Manhattan but three times that of Birmingham? If someone is led by the "small city" terminology to think that Wuxi is similar in size to Johnstown, PA (pop. 25000), or Wiesbaden (pop. 291000, metro area over half a million), they're already confused; wouldn't "googling population of Wuxi and comparing it to familiar population of city" clear that right up?
At least after you cut through the terminological ambiguity of whether "Wuxi" means Liangxi Qu, the five districts of Wuxi City, or the whole Wuxi prefecture, and the analogous ambiguity of where to draw the limits of "London" (the Square Mile?) or "Sydney" or whatever familiar city you're comparing to, it seems like you'd be less confused after making those comparisons, rather than more confused.