Yes, "Space Normal Speed" is accepted to mean 'Ahead 1/4 Impulse'. By this point in warp drive development, the term had become almost archaic in that most everyone just assumed that's the default sub-light speed (explaination below). In fact most Federation crews hadn't heard the term since their days at Starfleet Academy, hence Sulu's turning around in surprise when Kirk used that phrase. But you see, Kirk knew that the Commissioner (who was a politician and not a trained navigator) was still listening, and probably wouldn't know the meaning of 'Space Normal Speed'. It was Kirk's way of dragging his feet and pulling away from the planet reeeeaaallly slowwwwly, without getting yelled at more by the Commissioner.
One of the perks of service on a Starship is that a crewman gains about one year of extra life over the course of his career, as a result of the time dilation effect which occurs at high sub-light speeds (even the Big E Enterprise has to obey the bigger E Einstein

). To avoid accumulating excessive time dilation effects, most Starships keep their sub-light velocities to about one-quarter light speed, or 'Space Normal Speed' by default. If they want to go faster or farther for any length of time, they'll usually just go ahead and enter Warp, during which time the warp envelope shields the vessel from further time dilation effects. Upon exiting warp, one of the first things that the Starship's main computer does, is seek out the nearest Starbase's signal, and reset the ship's chronometer to compensate for any accrued time dilation effects.
This is one of many techno-details that had to be sacrificed by the writers, because it would either over-complicate the plot, or it wouldn't fit into the 45-minute timeframe allowed by the TV network. But thanks to resources such as this site, we can fill in the empty spaces.