While I agree that tweet as a noun is defensible, and is an exacting term, I wouldn't expect its use from professional journalists in a non-tech report without qualification about what the term refers to. It would be like using any other niche specific term without qualification. A good example would be etude, musicians and classical music enthusiests will immediately know it, but an average reader will not, even if the term is a similar term of art.
You can sometimes get around that with a brief inline gloss, like "Obama sent a tweet, as messages on the service are called, expressing his wish for...".
