What does the U.S. mean when it says it 'supports a strong dollar' ?

Second, when a current (or previous) administration official has voiced commitment to 'a strong dollar' – that's not what they mean, or what they've signaled to Wall Street.

What they mean is, or the way institutional investors interpret it is, 'There will be no actions or sudden policy changes that result in volatile moves in the dollar, higher or lower.' In other words, the 'strong dollar' commitment is code language that the U.S. will let the dollar weaken (or strengthen) slowly, according to market forces, as determined by fundamentals. The dollar has weakened about 80% versus the euro since 2000.

And right now, those fundamentals are dollar-bearish. To be sure, some progress has been made to decrease the U.S. trade deficit (the recession has reduced U.S. consumption of foreign goods, even as the weaker dollar as boosted U.S. exports). But much more work needs to be done by Congress to cut the U.S. budget deficit, and to ensure that the U.S. economy has sufficient demand to grow at an adequate rate.

Monetary policy also, obviously, influences the dollar's value: if real (inflation adjusted) U.S. interest rates rise above real rates on other, major currencies, that too, would support the dollar.

When those four factors line-up in favor of the U.S. (trade surplus, balanced budget, robust GDP growth, real yield), the dollar will strengthen versus the world's other, major currencies.