Four pages and everyone has missed the point.
The question of whether the United States is a 'republic' or a 'democracy', as it is popularly discussed today, appears to have originated in the mind of Karl Rove during the Bush administration. Similar to his attempt during the Bush years to switch the blue and red party colors of the Republican and Democratic Parties, the strategy was (as Merirosvo suggested) based on subliminal messaging.
The question of 'republic' vs. 'democracy' itself is rather silly. Democracy simply means 'government by the people' (Greek δῆμος/demos, 'people', and the suffix -κρατια/cracy, 'power'/'rule', i. e. government), as you will find in the first definition of the dictionary. Anything from communist one-party vanguard rule in the name of the proletariat to liberal representative democracy by secret and universal suffrage sees itself as a 'government of the people'.
'Republic' was once used in a similar way to 'commonwealth' and could literally refer to any state, any government or even a community striving for the common good. The word retains a similarly positive meaning in common parlance today, except that monarchies are now excluded from the definition. This isn't to say that monarchies can't reflect the popular will. Often monarchs, including hereditary monarchs, don't base their authority on divine right but rather on an elective basis (in elective monarchies), vague popular notions (think 'King of the Belgians') or ratification by the legislature (ancient Germanic traditions still present in Spain).