Here's how I define '00s in terms of "old-schoolness".
2000: Very, very old school. Just slightly more new school than 1999.
2001: The first 2/3 of the year was a lot like 1999-2000, but Fall 2001 was what really ushered in the early '00s "9/11" era.
2002: Definitely '00s, but still very old school. Aside from terrorism fear, very little remains of this time, and the technology, TV and Web looks quite similar to that of the late '90s. This was also the last year that was not really a part of social media movement.
2003: The peak of the "Early Noughties". Totally '00s, but somehow it's still more like 2000 than 2006. Quite old school. The very beginning of social media movement.
2004: The last year that is really 100% "back in the day" relative to 2016. Some very primitive late '00s/early '10s things were already present by then, but it's still just before the "Age of Tech" started to immerse everything in the late '00s and early '10s.
2005: Old school, but a little less old school. Crunk rap gave into snap rap, XBOX 360 and YouTube were introduced, but it's still more like 2003 than 2007.
2006: The last year that's really a different era from now (makes sense, since it's 10 years ago already). Emo was at its prime, a shift began in fashion, music and tech, and social media became pretty ubiquitous. Basically a border between Old and New School. Still it's more old school than 2007-2009.
2007: Very '00s, but still new school from a 2016 perspective. Nothing REALLY retro about 2007 yet.
2008: Noticeably dated, but still quite new school. This was when the Great Recession happened, the second social media boom began to take place, and popular culture, music and technology went through another big shift yet again.
2009: Definitely new school. The very beginning of the current era, arguably the beginning of the current cultural decade. A lot of things introduced this year are directly connected to the early '10s (but not so much mid '10s), though there was a plenty of '00s stuff that was still going strong by then.