[From my comments in the rebranding survey]
Wikimedia vs. Wikipedia
Our overall community centers around the current Wikimedia concept, not Wikipedia. Naming the whole from one its parts is ambiguous, confusing, and disrespectful to non-Wikipedia projects. The majority of the population of the United States is white, but it would be absolutely preposterous to rename the country to the White United States of America, even if that is how people in other countries (and Americans) think of it.
We are not selling a product or service. I think it is _good_ that some organizations and people do not know about our plethora of projects, as that gives us an opportunity to talk with them about the other projects. I believe that changing the name to "Wikipedia" will make it more difficult to get outsiders to pay attention to non-Wikipedia projects.
I believe that moving to "Wikipedia" will damage our reputation. In addition to the reasons above, it will likely alienate at least some of those involved in non-Wikipedia projects. It could turn the community into the Wikipedia community, as our other projects fade away.
Paul Weiss User:Libcub