How do I list third party contracting positions on a resume?

The global, major well known main company has their laboratory IT work contracted out to a global outsourcing company. The global outsourcing company has hired me to be lead consultant for an upgrade project. So the work I am doing is for the main company. I want to update my resume to include this position, how do I list it on my resume? Also, since I'm one of the only people that has done this upgrade before, I have no job description beyond "Lead Consultant for" the project. I'm the extra resource with relevant experience. So when I list it on my resume I need to write my job duties as a combination of what I've done in the short time I've had the position, what I'm doing, and what I know needs to / will be done. Any advice in regards to that would be appreciated. In the past I've listed contract work as "'Main Company (contract through 'contract agency')", but that was all straightforward working in the lab contract positions. Now I'm doing consulting work. The story: Main company eliminated all of their in house expertise and then hired back some percentage of those affected on a contract basis through a global outsourcing company. This made a mess of their needed upgrade project so they hired me because I'm one of the few people that has experience doing the upgrade. Basically a recruiter got a hold of me and hooked up the position. The recruiter is CEO of an unknown S corp that serves no function other than making me aware of the position and doing the taxes that are the difference between a W-2 (which I am) and a 1099. But that recruiter's S corp is my actual paycheck paying employer. I'm thinking to put it on my resume as one of the following:

Main Company (Lead consultant for X project)
Main Company (3rd party consultant)
Outsourcing Company (as outside consultant, for Main Company (project X?))

Or some variation of those. Any advice on which is the "correct" or at least best and accepted way to list this on my resume?