anxious. Since anxious comes from anxiety, it should contain some connotation of being worried or fearful and not merely eager or expectant. You may be anxious to put some unpleasant task behind you, but, unless you have invested money in it, you are unlikely to be anxious to see a new play.
The rule here is that anxious should not mean "eager". However, the only evidence Bryson supplies is etymological: "Since anxious comes from anxiety, it should contain some connotation of being worried or fearful". This is the etymological fallacy: the belief that a word's history determines its meaning. And the history that Bryson gives is wrong. The OED on anxious:
f. L. anxi-us troubled in mind (f. ang-ĕre to choke, distress) + -OUS.
The OED on anxiety:
ad[aptation of] L. anxietāt-em, n. of quality f. anxi-us: see ANXIOUS, and -TY.
So anxious is not derived from anxiety; both words are borrowed from different Latin words.
On the rule itself, Bryson is in good company: the rule that anxious should not mean "eager or expectant" is repeated in many usage books, and Merriam-Webster's Dictionary of English Usage tells us that it is a shibboleth in American usage. On the other hand, it has meant "Full of desire and endeavour; solicitous; earnestly desirous (to effect some purpose)" since 1742. Merriam-Webster's Dictionary of English Usage quotes Byron, Carroll, Dickens, Darwin, Thoreau, Kipling and Flann O'Brien all using anxious in this sense.