This has been explained actually, if I can find the post but I remember what was said.
Basically the system calculates EVERY single piece of gear your character has gotten regardless of your CURRENT overall item level from the gear you have equipped (EG if at some point you had got a 360 item but your 355 was still better statwise, the system will calculate that slot at 360 because you obtained the item regardless).
So this is why people are seeing some obtained their 370 Emissary at 361 and others not until 365~ average.
Conclusion could be the people who receiving it quicker than those at 361/365 currently have been lucky with Titanforges or Warforges.
Either way the cut off item level for it is 365 (whether your character has it equipped or not).