First and foremost, the term "Red Zone" is not an official stat. It has origin stories that originate from either a former head coach, or a broadcast director depending on who you ask Source of Red Zone.
The definition of a team "in the red zone" is that they ran a play within 20 yards of the goal line (a touchdown).
When teams are positioned anywhere within 20 yards of the opponent's goal line, they are said to be in the red zone, a place on the field offering high-percentage scoring potential. The Complete Idiot's Guide to Understanding Football
Therefore in your example, any play snapped from a spot inside 20 yards (regardless of which down it is) would be considered a red zone play and count as a red zone visit for the team.
However, there is some statistics that point to a lack of correlation between red zone efficiency and total offensive performance:
On average, NFL teams ran 132 plays in the red zone and converted just over 20 percent of said plays into touchdowns, for a league average of about 26 red zone touchdowns per team. Overall, there was a .595 correlation between red zone touchdown percentage and offensive scoring, which should come as no surprise to most football fans. Scoring touchdowns as opposed to field goals in the red zone will lead to more points, but there are obvious outliers such as big plays and long touchdowns. A .595 correlation isn't outrageous by any means (if it had been 0.7 or higher we'd really be in business), but it's positive enough to draw the rather obvious conclusion that it'd behoove us to lean toward drafting players who could have increased opportunities in the red zone in more efficient offenses. Quote Source