You're mainly correct about the point on transition. Without getting too much into the weeds of oceanographic theory, since El Niño/La Niña phenomena remain somewhat complex events, a short and skinny of Nino 3.4's origins as a metric for defining these events is explained by the topographic nature of the equatorial Pacific, functions of the trade winds, mainly. To the west (Nino 4), the Pacific Ocean has a deep mixing layer (thermocline), such that, say, the isotherm at 25 degrees C is deeper at the Solomon Islands, while the thermocline is much shallower in the east off of Peru (Nino 1+2).
The trades push against the western Pacific and an accumulation of warm water (the "warm pool") and higher sea level is formed naturally, while the eastern Pacific is shallower and cooler. Warm water upwells in the west; cool water upwells in the east. This temperature difference drives the Walker Circulation, which typifies weather patterns in the Pacific and ultimately affects weather in much of the world. The stronger the pressure gradient and trade winds, the stronger the Walker Circulation, and this usually is associated with La Niña. Because of imbalances in the wind stress, however, this circulation can (and often does) break down, and the warm water in the western Pacific moves eastward towards the South American coast. The thermocline therefore deepens in the east, driving the positive feedback that causes trade winds to weaken, causing water to warm further, which weakens winds still. El Niño. Introducing ocean currents to this function would make this entirely too lengthy, so I'll stop there.
It is said that Nino 3.4 is kind of the "sweet spot" where the atmospheric-oceanic teleconnection takes place, where it is closer to the warm pool than Nino 3, and convection appears during El Niño. Kevin Trenberth has noted that other metrics, such as the Southern Oscillation Index (SOI) and Nino 3 have been used before to define El Niño/La Niña, but their shortcomings were supplanted by the Nino 3.4 metric that has been largely adopted.
The paper that is germane to this is Trenberth 1997, "The Definition of El Niño"LINK
Nino regions 3 and 1+2 have been considered important in recent years, due to the discovery of new El Niño and La Niña events called Modiki El Niño and Modiki La Niña, where Nino 3.4 is warmer (cooler) than average, while surrounding regions are cooler (warmer) than average. This has somewhat different atmospheric effects.