"Gene's Vision" has been an effective marketing strategy for the franchise since the seventies, but as many others have posted, it doesn't have much concrete meaning because it (a) exaggerates Roddenberry's status as a "visionary," (b) gives him credit for the creations of others, and (c) has been constantly revised to sell whatever the latest product happens to be.
When Star Trek: Voyager
came out, the idea that Roddenberry at first wanted a woman as the lead in the original series was fed to the media more than once. Earlier, Brandon Tartikoff wanted the dedication in front of Star Trek VI: The Undiscovered Country
to invoke "Gene's vision." Roddenberry, of course, had almost nothing to do with that film, clashing in meetings with Nick Meyer and ultimately refusing credit as "executive consultant" before his death. And so on and so forth...
Anyone decrying the fact that the Bad Robot version of the franchise conceives of it as an action-adventure series has obviously never read a Roddenberry story memo from the 60s. Rarely did an episode go by where Roddenberry didn't comment on ways of increasing the action-adventure potential.