The fireteam is a small unit with decentralized leadership under the command of a non-commissioned officer or senior non-rate built around the firepower of a squad automatic weapon. At the time of it's inception, squads were organized around a single SAW. Under this organization, each fireteam consisting of four men would carry a SAW, allowing each unit to suppress and move. Key points here: Decentralized leadership, smaller and more mobile force, NCO in charge until he gets killed, and a squad automatic weapon.
Let's have a look at your previous examples and see if they fit this description.
The Phalanx: The Phalanx is a formation, not an element. Its size varied, depending on the number of men available. This would be like saying a wedge is a fireteam. A platoon can assume wedge. A company, even. Hell, if we wanted to, we could get the whole damn battalion out here to jump in a wedge right quick. It seems a bit stupid, but it could be done.
Han Dynasty: The smallest unit organization during the time of the Han Dynasty was a platoon. A platoon is not decentralized leadership, it's not a small unit, it's definitely not the most mobile element. That's not even considering the fact that the element wasn't capable of levying mobile suppressing fires, and definitely didn't engage in fire-and-maneuver, or fire-and-movement tactics. They didn't have non-commissioned officers, and in fact, they had two sides to their army. Non-professional (conscripts) and Professional (volunteer). Professional is about as close as they get. No NCO, no decentralized leadership, this isn't a fireteam.
The Revolutionary War did feature guerrilla warfare style tactics, but they did not feature professional military men organized into small teams, lead by an NCO. Not fireteams, just pockets of fighters. I mean damn, the French had to show us what the bayonet was for.
How did the Russians fight the Germans? According to their organization, in squads. Squad size of the day was 8 men, and I'm talking in particular about the SMG platoons, because this is about as close as we get. 3 squads to a platoon. No NCO leadership over those squads. Furthermore, the fireteam is a breakdown of the rifle squad, so this example is further disqualified.
What I'm telling you -is- the "official" version, as I'm a Marine infantryman of 8 years.
Your main arguments so far have been that a platoon is a fireteam, a squad is a fireteam, a formation is a fireteam, and more recently that the Army Rangers have a 7 man fireteam. The reality is that a platoon is a platoon, a squad is a squad, a formation isn't an element, and the Army uses the same 4-man set up as us in their fireteams. On occasion, we'll form what we call a fireteam reinforced, where stragglers from a hit element will form up on an existing fireteam. Only then would you see something on the level of a seven man team. Ther's a definition for these things, there's a reason for the definition. These specifics become kinda important when they're your job.
Finally, I'm not sure what's up with the snide "geek out" remark, or the neat little eye rolling smiley figure over there. We can disagree on something without acting like kids.
