I generally remember web 2.0 as being about social, not exactly about open APIs. In fact, curious I Googled the term to see what the Wikipedia article has to say and sure enough, the first defining paragraph reads as follows:
"A Web 2.0 site may allow users to interact and collaborate with each other in a social media dialogue as creators of user-generated content in a virtual community, in contrast to Web sites where people are limited to the passive viewing of content. Examples of Web 2.0 include social networking sites, blogs, wikis, folksonomies, video sharing sites, hosted services, Web applications, and mashups.[3]"
Certainly a big component of web 2.0 was mashups, and for those you need an open API ... but I'm not sure it was the defining characteristic.
"A Web 2.0 site may allow users to interact and collaborate with each other in a social media dialogue as creators of user-generated content in a virtual community, in contrast to Web sites where people are limited to the passive viewing of content. Examples of Web 2.0 include social networking sites, blogs, wikis, folksonomies, video sharing sites, hosted services, Web applications, and mashups.[3]"
Certainly a big component of web 2.0 was mashups, and for those you need an open API ... but I'm not sure it was the defining characteristic.