Which is the "Action Phase"?
You're right that the Action Phase is unique to the Brood Wars Expansion. A quick search of the phrase "Action Phase" in the digital version of both rule books only turned up results in the Brood Wars version. 

Like you, I found a lot of people saying that the Action Phase was synonymous with the Execution Phase (example here). The most convincing piece of evidence is straight from the Fantasy Flight Games forums. This question is answered by saying that the Action Phase is identical to the Execution Phase, but since it's on the game designer's website, I would imagine a moderator would have corrected it had it been wrong. It looks like questions are answered by people from FFG at least occasionally, so they are probably looking in on rules questions to make sure information was correct. Not the solid answer you were looking for, but I think it confirms the two are the same.