Older members of WikiWikiWeb (see MissingWikiBeforeXp) have referred to it as "WikiWikiWeb", "WikiWiki" or "Wiki", all expressions without the definite article "the".
Some citations of WardCunningham that contain the term "WikiWikiWeb" (referring to this WikiForum) as a head instead of a modifier:
As suggested by the earliest version of the page called "WikiWikiWeb" available on the Internet Archive WaybackMachine at http://web.archive.org/web/19961129194919/http://c2.com/cgi/wiki?WikiWikiWeb, the expression "WikiWikiWeb" originally referred to the PerlLanguage WikiEngine (http://c2.com/cgi/wiki) together with the WikiForum it runs.
How is that suggested? It's not stated explicitly.
Right. It's not stated explicitly, but they use it that way: They say "WikiWikiWeb" instead of "the WikiWikiWeb". Originally, there was no distinction between the WikiEngine and the WikiForum, because they were both regarded as a unit (called "WikiWikiWeb", without the definite article). Later, on the page "WikiWikiGoesPublic", WardCunningham announced WikiBase, the second WikiForum running on the WikiEngine, and from which a variant of the WikiEngine itself could be downloaded automatically through the HyperPerl generator. So since the WikiEngine wasn't unique to this WikiForum anymore, people began to make a distinction between the WikiEngine that runs this WikiForum and the WikiForum itself (see WikiHasManyMeanings). Somewhat confusingly, they are both called "WikiWikiWeb", both without "the".
Many newer members (see HeatDeathOfWiki), however, find using the expression "WikiWikiWeb" without the definite article shocking, so they prefer putting the definite article "the" in front of "WikiWikiWeb".
Question: Why do new members of WikiWikiWeb tend to use the expression "the WikiWikiWeb" with the definite article "the", and which people tend to say "the WikiWikiWeb" instead of "WikiWikiWeb"?
Perhaps because it just trips off the tongue better. It's not just newbies that do this. The trend for some years has been towards including the definite article.