Meh, this is the worst kind of pedantry imo. Like, first of all, doesn't matter. But if you must, you're going to have to go with 'as would be defined by mathematicians' - because they spend a lot of time thinking about these things, and thus are by far the most qualified to have an opinion....
And the mathematicians have thought long and hard about how the Euclidean concept of a straight line generalizes to other geometries... and came up with geodesics... aka great circles...
Why would mathematicians take precedence over people who navigate ships and airplanes for a living? Also, ellipsoidal geodesics aren't always circles.
"Thinking long and hard" isn't actually much of a qualification, if you think about it long enough and hard enough. Playing around with the definition of "straight line" is just an amusement, putting different theoretical constraints on the recreational problem. The whole thing is pedantry to begin with, so don't be surprised when someone pops their head in with something unexpected just to show off how clever they think they are.
Launching pointless academic arguments is almost the whole point. It shows the audience that everyone involved is very smart, and all possibilities have been duly considered, and therefore the agreed-upon answer must be very significant, reliable, and noteworthy.
If you navigate a ship while keeping the rudder straight, absent current or wind you will be following a geodesic. If you try to follow a parallel you will have to constantly be turning.
Planes follow geodesics too over oceans, not parallels.
Yes the geodesic on an ellipsoids aren't always great circle, but the earth's geodesics are commonly referred to as "great circles" because the earth is very nearly spherical.
Should we ignore wind and current? If you are in a medium that has winds and currents, and you do not touch your yaw controls, how will you know if your course remains on a straight line?
You either adjust your compass bearing (also correcting for magnetic variation), or you adjust course.
And the mathematicians have thought long and hard about how the Euclidean concept of a straight line generalizes to other geometries... and came up with geodesics... aka great circles...