You read it right, it just glosses over the speed of the relative reactions and I should have been more clear/exact with my words. You can oxidize benz carbons with various oxidants but you have to hit it much harder than you do to oxidize an alcohol. To oxidize your alchohol with Jones conditions you will be running at -10 to 0 ºC for about 30min. To oxidize your carbon you will need to heat it for an extended period of time.
I have run this reaction many times on a variety of substrates with great success unless my compound turns out to be unstable to the strongly acidic conditions. A good quick guide can be found on notvoodoo. Chromium salts are poisonous, so show them the respect they deserve and you won't have any problems.