I don't think these things are mutually exclusive.
Twitter didn't really know what problem they were solving. They may be a bit of a special case. But they built something they felt was usable and launched it.
If you look at the Twitter launch at SXSW it was all about where people were going and how the band was. It fit very well into that model.
At the time, twitter didn't have search, they didn't even build their own search, somebody else did (summize) and Twitter bought them. They didn't know about trending, they didn't know that business would make up the bulk of their publishers (my suspicion, correct me if I'm wrong). They didn't know that the majority of their revenue would come from selling access to the firehose of real-time data.
So, you have to build for the now, and you have to respond to the future.
Facebook wouldn't be what it is today if they had just built for their 'now' which was initially putting the college facebook online. They did that, and kept iterating for the future.