As long as you're buying bitcoins with dollars, I can't see how anyone could be sure they're completely anonymous.
Upgrade your sources of information. Nobody who understands how bitcoin works makes the claim that it's anonymous. Pseudonymous, yes.
And, with difficulty and effort, they can be made reasonably anonymous - but never perfectly. Your example is valid - and any time you link your name to a bitcoin transaction, that name and that wallet are pretty much linked forever: so it follows that if you stay in bitcoin exclusively, never link a name, never buy anything in meatspace that requires a meatspace name and address to ship... etc., you'll be sufficiently anonymous for most practical purposes. The are also coin mixers that will launder your existing coins, disassociating them from your current wallets.
Have fun with your research into this well-documented issue!