Yes, there are different definitions on both sides of the straight. And while the independence side has gained strength in recent years, the official stance has been that China and Taiwan are a part of one country, however defined, and, to my knowledge, no official government act of the ROC has been promulgated to the contrary.
As I stated in my original reply, independence isn’t a cut and dry issue. To assert that these claims are “dusty” historical claims in light of what you’ve written here is disingenuous.
Edit: I should add that there is only one government in Taiwan, the ROC. The stances of the individual political parties cannot be taken as synonymous with the stance of the ROC itself. To do so would be as ludicrous to say that there’s no right to an abortion in the United States, because 1 of the two major political parties opposes it. The most that can be said is that both abortion in the US and independence in Taiwan are controversial and delicate political issues within each respective jurisdiction.
The 1999 "Resolution on Taiwan's Future," since elevated to DPP party platform, is crystal clear on "two Chinas."
> it’s official government policy on both sides of the straight that there’s “one China”
That's an oversimplification. The KMT / Pan-Blue believe there's "one China" in the sense that they believe the PRC is an illegitimate government, currently occupying mainland Chinese territory that belongs to the ROC.
The DPP / Pan-Green believe there are two Chinas, with Taiwan as an independent entity from the mainland PRC. Their current official position is that this state of affairs already exists, and therefore there is no need to make any declarations or changes from the status quo.
So in summary, about one half of Taiwanese political power considers the communist government as illegitimate, and the other half considers Taiwan independent. Which I would guess is fairly different from what the PRC defines "one China" as.