It depends in large part on the precise nature of your icosahedral projection. On your first image, you show two rather different icosahedral views: the "Traveller" hexagonal one that is designed to get the idea across, but isn't really a nicely defined projection as such; and a nice icosahedral one that looks to be twenty piecewise stereographic projections arranged into a generally similar shape. The vertical split on that last face on the Traveller map is designed to make things fit more nicely on a rectangular page, but it's a little annoying to calculate. Fractal Terrains ( http://www.profantasy.com ) can export the Traveller-style thing using its Cosmographer template feature ( File>>Export World>>Cosmographer Template ), but it can't go the other way.
Wilbur ( http://fracterra.com/software.html ) has had a piecewise linear transform that converts an Equirectangular map into something that folds into an icosahedron for a couple of decades now using the Texture>>Other Maps>>Icosahedral Projection menu option. It also goes the other way if you hold down the Shift key when using the Texture>>Other Maps>>Icosahedral Projection menu option. To a good approximation, it's an interrupted Collignon projection.
It looks like your conversion is using a piecewise Stereographic (or some related projection with curved parallels) projection to convert the Traveller map to Equirectangular. This kind of conversion generates unexpected results if you (like too many people I've asked over the years) assume that those straight lines between segments are lines of latitude, which would make the whole thing some kind of projection in the cylindrical family. The conversion that you're showing here converts those straight lines along the Traveller map segments into curves on the flat map rather than keeping them straight, which will only matter for specific uses of the map. The hexagons are also distorted on the converted flat map, which may be the exact right thing or it might be bad, depending on the player in question.
I guess it all comes down to deciding what the Traveller map means. I've seen a folks get out a ruler and measure things on the hex maps, for the couple of times that they wanted a distance. It's a tough thing to answer, because the maps are generally so coarse and notional that the distortions map no real difference in accuracy for linear or non-linear.
And yes, it does look like I've completed my transition to crusty old curmudgeon.
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