I thought that the Vasari came from the TEC since they were outcasts of the TEC. Nevertheless, in both the manual and in the game, it did say it is a Vasari who developed first, so naturally it would cost more labs to re-engineer foriegn techs

Like Kitkun said, Advent came from the Humans.  So, its more like this, Advent started out when TEC was still known as the Trade Order and they were cast out by them and they come back 1000 years later. 

Anyways, in the manual (and I'm summarizing here), the Vasari encountered these devices and they don't know how they work but they were able to replicated it with their nano tech.  So, they get them first.
The TEC encountered these devices and knew it wasn't of Vasari design and are unable to reverse engineer it and they also don't know why/how it works and so the only ones being used are the ones they stole from the Vasari.
Now the Advent started to notice these when they started getting closer to TEC central worlds and now are trying to get as many as they can from captured TEC worlds.

So, my point is that Vasari are the only ones making them so it makes sense for them to be able to get them the quickest but the Advent are getting them from the TEC, so they should get them last, not second.  Pretty much, first should be vasari, second tec and third advent.

Yeah, I thought of it that way for a bit but then I reasoned that the TEC have had much more contact with Vasari, therefore probably stole more in those 10 years and Advent are just starting...in either case, I think it should either be even for Advent and TEC or TEC ahead to follow the story.

If they have to be stolen, shouldn't the other races be blocked until any Vasari research the device?  Weapons that were slightly different and more consistent with the technology of other races might be an improvement- how about a 'Circe' device that affected only a portion of a fleet, but for longer, and some kind of ECM device that has a reduced effect, but also a slight negative modifier on enemy combat?

How about some kind of game mechanic for 'theft' of research- you could get a free research of your own for every three-five enemy labs you destroy?  Perhaps each race could have a few types of technology that could be stolen?  It could be an alternative target for raids behind enemy lines.  If races had some technologies that could be stolen, the top level inhibitor, requiring some stolen technology, could combine the basic one with ECM and advanced shielding. 

The greatest problem with phase jump inhibitors seems to be that the AI don't prioritise hostile ones sufficiently before attempting a retreat.