3rd Edition is 3rd Edition because it replaced 2nd Edition. It very easily could've been called Advanced D&D 3rd Edition, but WotC determined that calling the game "Advanced" was bad marketing and duly dropped the new-player-repellant from the title.
There wasn't a reconvergence of the twin game lines; the Original/Basic D&D line was killed off dead. Its last release was in 1996; WotC acquired TSR in 1997; and in 1999, WotC's TSR released the
D&D Adventure Game as an introductory product
for AD&D 2nd Edition. An AD&D boxed set wearing the D&D brand name, a year before the release of 3e.
When I first got my 3e PHB in 2000, I remember being struck by the clear continuity between 2e and 3e in at least one interesting place: spell descriptions.
Read the text of fireball as it appears in 1e, 2e, and 3e, and you'll see the sort of thing I mean.
