Isn't Web 2.0 the web with user-contributed content, i.e. social media? For me, being part of Web 2.0 would not be something to be proud about, it's the tech that brought uninformed, angry populist rage online...
I think Web 2.0 was more when the web became more interactive. Instead of only static web pages there were forums, UIs that responded to your actions, and early social networks (but not the modern monsters we currently have). It enabled them, but it wasn't strictly "Web 2.0".
That said, even at the time the definition was not really clear. I remember a joke, why Web 2.0 is like teen sex... Everyone is talking about it, almost noone is doing it, and even those that are, are probably doing it wrong.
Fun times. You could change the image without reloading the page. Woohoo!