So let's say I wanted to kill, for example, David Connor with the death note. But his full name would be David Winston Connor. Must I write his full name, or is David Connor enough?
David Connnor should be enough, in How to Use I it says
This note will not take effect unless the writer has the person's face in their mind when writing his/her name. Therefore, people sharing the same name will not be affected.
That alone should be enough. if, by some genetic miracle, 2 identical people with the same name existed then adding the person's full name (ie. Troy Delu vi La'Flash) may then allow the Death Note tell the difference between the 2.
IF, and this is a huge if, that 2 identical people with identical full names existed then it's quite possible they will be immune unless the cause of death was
David Winston Connor dies from a heart attack in 40 seconds after writing his own name in the death note
but try and get someone to do that, also the odds of 2 people like that being born is insanely huge.
NOTE: To clarify a point raised by trlkly, How to Use: LIII talks about writing one's name across multiple pages, Point 1 says it wont work but Point 2 says that the front and back are still one page and talks about writing the surname on the front and given name in the back. this seems to imply that by name it would refer to the given and surname.
I think David Connor should be enough as the death note requires you to know the name of the person AND know the face of the person. So as long as you have the face in your head, the middle name is not important.