> When you go fast, he said, the world around you changes shape, and time outside starts moving slower.
> Since the boats are going fast, the space doctor’s special idea says that their watches will run a little slower than the ones on Earth.
Shouldn't time outside start moving faster in the first quote?
When you go fast, and look outside, it looks like it is the outside that is moving fast (unless you are experiencing acceleration, at which point General Relativity kicks in). Therefore, to you, it looks like time outside is moving slower.
When the space boats are doing their thing, to us on the outside it looks like they are moving fast. Therefore, to us, it looks like time for them is moving slower.
There is no contradiction because events with identical x-positions do not have identical X-positions.
If you already know what a partial derivative is, see the first paragraph of this: http://mathpages.com/rr/s4-07/4-07.htm
If I'm moving very fast with respect to you, I see your watch running slow, and you see my watch running slow.
So the first paragraph is true, if it's talking about your view - the world around you changes shape, and time outside moves slower, as seen by you.
The second paragraph is true, as seen from Earth, not from the boats.