The following article contains an answer for the question: What is the name of the song for which Garbage's first mix was built upon in the final recording made in August at Armoury Studios in Vancouver, Canada? , can you please find it? 

The first recordings were made during the European leg of Garbage's Version 2.0 world tour. After listening to the orchestral demo, the band worked on the key and tempo. Garbage used a portable studio from a number of European cities to record material for Arnold, keeping in touch by phone as he produced the song's string arrangement in London. Since the strings carried the structure of the song, they had to be finalised and recorded before Manson could sing her parts. Arnold recorded the strings with a 60-piece orchestra in one day at London's Metropolis Studios.
Garbage flew to London for a day to record the basic tracks, laying down electric guitar, bass guitar and Manson's vocals with the orchestra. Manson called working with the orchestra "exhilarating". That night, the band flew to Switzerland to resume their tour for three weeks. 
The final recording was made in August at Armoury Studios in Vancouver, Canada, where Garbage built upon their first mix of the song, adding and subtracting parts, and completed final recording and mixing. The band kept the arrangement tight to preserve the song's dynamic, sweeping melody. "The orchestra took up so much space and really dictated where the song was going dynamically," keeping the recording simple, Vig recalled. "Besides the drums and bass and some percussive loops, there's a little bit of guitar that Duke and Steve did. There's not a lot of miscellaneous tracks on there. There's a few little ear-candy things that we did, but it's all meant to work around Shirley's singing." Although Garbage owned its own recording studio in Madison, Wisconsin, for legal reasons the song could not be recorded in a U.S. studio. "The World Is Not Enough" was completed, mixed and mastered at the end of the month, and the group returned to their recording studio in Madison to record their mix of the song. Garbage's version (the "chilled-out remix") downplayed the classic Bond sound in favour of the band's style. Vig later said about the original recording, "We're pretty pleased...

The World Is Not Enough