I don’t know if anyone is credited with creating the word “Tweet” but it seems to have been synonymous with Twitter since the start. So many Twitter scripts and Twitter related websites use the word “Tweet” in their domain name because of this, including this one.
Twitter is obviously aware of this so they have taken steps to trademark the word :
We have applied to trademark Tweet because it is clearly attached to Twitter from a brand perspective but we have no intention of “going after” the wonderful applications and services that use the word in their name when associated with Twitter. In fact, we encourage the use of the word Tweet. However, if we come across a confusing or damaging project, the recourse to act responsibly to protect both users and our brand is important.
So it seems that they are happy for people to use the word “Tweet” on their websites, as long as it is used in a responsible manner and isn’t hurting the Twitter brand in any way.
Link : May The Tweets Be With You