Not on "Like a Rolling Stone", you're correct. Two different stories, two different sessions---both happened. Robertson had already been on the Bringing It All Back Home album (the "electric" side, of course), and would be on Blonde on Blonde the following year (credited as Jaime. He is listed as J.R. Robertson on Music From Big Pink), along with fellow Hawks Levon Helm and Garth Hudson.
And it wasn't just Robbie and Levon who were hired for the '66 tour, it was the full Hawks, including bassist Rick Danko, pianist Richard Manuel, and organist Garth Hudson. Levon didn't care for getting booed by Dylan's diehard Folk fans, and left the tour (he ended up working on an oil rig in the Gulf, until he got a call from Danko in Woodstock, telling him to come on up. The rest is history!), replaced by Johnny River's drummer Mickey Jones.
Dylan is credited as "discovering" The Hawks/Band, but it was actually Albert Grossman's secretary, who had seen them on the Jersey Shore playing in a bar. Albert was John Hammond Jr's manager, and knowing Hammond was looking for a band, put he and The Hawks together. They are his band on the So Many Roads album, and were his touring band in'65. Dylan heard them, and hired them away from John.
Kooper's telling of how he ended up being on "Like A Rolling Stone" is great story telling. It started as Onhwy61 says---hearing Bloomfield warming up, Kooper realized he was severely outclassed, and mosied over to the studio's Hammond organ. He started following along, playing along with the others. The reason he waits half a beat on every chord change in "Like A Rolling Stone" to play the chord is that he hadn't heard the song before, and didn't have the chord chart. He had to wait until everyone else played the changes, to hear the next chord!
