The most reasonable interpretation of this is to follow the latitudenal geodesic along its eastern path. You cannot claim that one geodesic is more “straight” than another in 3d Euclidean geometry, that is nonsense. But that is what the author does.
Edit: Ok, the latitudinal geodesic only exists at the equator, so the question is fundamentally impossible, with how the author defines a straight line.
There is no such thing. A curve of constant latitude on Earth, except for the equator, is not a geodesic.
> You cannot claim that one geodesic is more “straight” than another in 3d Euclidean geometry
In terms of 3D Euclidean geometry, neither a curve of constant latitude on Earth's surface nor a great circle on Earth's surface is a straight line/geodesic. Both are curved.
If you restrict to the 2D surface of the Earth, a great circle is a geodesic but a curve of constant latitude, except for the equator, is not.
You're right, so if the puzzle restricts one (arbitrarily) to geodesics as the only valid straight-lined curve (?), it becomes nonsense because the only easterly geodesic is at the equator, and seattle is not at the equator.
Wrong. There is a perfectly good great circle passing through Seattle and pointing due east at the point where it passes through Seattle. The author showed it to you in the article. That geodesic does not always point due east, but nothing in the article said it had to. The article only said you have to face due east at the start, not that you need to continue facing that way.
> the original instruction was "straight line", not "geodesic"
If you're working within a 2-sphere, such as the Earth's surface, or indeed any non-Euclidean geometry, they mean the same thing. More precisely, there are no "straight lines" in the exact sense you mean in a non-Euclidean geometry, but there are geodesics that satisfy all of the geometric properties of "straight lines" within that non-Euclidean geometry.
> Imagine you begin a journey in Seattle WA, facing exactly due east. Then start traveling forward, in a straight line along the Earth’s surface.
where there isn't any wording saying "go east in a straight line".