I think there are some distinctions that should be made.
The first one harks back to the original question about being unique versus trying to imitate others. "Unique" is what the brand/brand experience should be. Some of the obvious ways that you can express the uniqueness are in the visual design (logotype, color palette, etc.) and in the marketing copy.
"Similar" is what the "information design" should be. By information, I mean the interface, the general design of the functionality/interaction, size of fonts, etc. This is for two reasons. The first is because as users become accustomed to performing a function in a certain way and that a certain button looks the way it does, it becomes important to design your interface following that convention because it improves "usability" and your users don't have to think about how to perform an operation.
The second reason stems more from contemporary graphic design theory (circa 1950s!), primarily the "Swiss school" in that information should be designed in a "neutral" way. This means that you don't want to draw attention to the design itself but rather you should draw attention to the content. The "design" should go unnoticed by the reader, viewer, user, etc. The font you use should not distract the reader from reading the text. So we use Arial or Helvetica and not Comic Sans (to cite an obvious example). This explains a lot about the design of a newspaper like the New York Times, for example.
I think this is the distinction you are noticing. I downloaded Instagram because I was not familiar with the app. What I notice is precisely this philosophy as applied to mobile design. The photos, the photo galleries, the single column of text all make full use of the medium. So the designers are designing for functionality and adapting the design to the medium/context. Because mobile screens are so small, it makes sense that the photos would be nearly full screen (on my iPhone) and that there would only be a single text column. There isn't much room (screen resolution) for anything else. More importantly, the app is about sharing pictures so that is what is prominent about the design.
The unique aspect of the app is the branding, in this case the logo. The rest of the information design imitates conventions elsewhere: the button design, the sizes of the fonts, the horizontal navigation bar etc.
Now if you look at the landing pages for the apps like Instagram they look very "Web 2.0." Again, I think this simply goes back to "designing for the medium." Landing page tend to be designed for computer screens like monitors and laptops, hence they have a lot more white space. Branding also is more important and so the logos are bigger, the brand's color palette is more pronounced, etc.
So rather than trying to distinguish between Web 2.0 and 3.0, I think what you are noticing is simply the application of "modernist" design principles to the mobile context.
As far as who started what, my guess is that Apple is also a, if not "the," leader in app design. My guess is that most app designers imitate the iPhone design and functionality.
Recently, I noticed a change to Facebook that reminds of what you are calling Web 3.0: now when you look at someone's photos a big black screen appears with very large photos that you can scroll through. This effect is called a "lightbox" effect but the visual design and mostly the size of the photos makes me think that the Facebook designers are taking cues from mobile design.
I hope this long-winded answer helps.