On the "Neptune and back in six minutes" thing, there's too much uncertainty to tell whether it matches the TOS or the TNG scale for low warp factors.
Assuming they started from Earth, the distance is somewhere around (4.5 billion klicks plus to minus 150 million klicks) times two, that is, 8.7 to 9.3 billion klicks. This in six times sixty seconds is about 24 to 26 million km/s, or 81 to 86 times lightspeed. And 85 c is just around warp 4 in both the supposed TOS equation and
the TNG Tech Manual graph. 30 million km/s would be 100 c, which again hovers between warp 4 and 5 in both scales.
It might be fun to think that TPTB in charge of "Broken Bow" or ENT in general used the TOS scale (even though TOS never used the TOS scale!), but it's just as probable that they used the TNG scale which would have been readily available to them in tabulated form in the Encyclopedia
, without the need for any hands-on math.