It was Kevin 'Caveman' Shirley who put Geddy back onto the jazz bass before the recording of Counterparts. I had a book with a part in it where Geddy talked about putting tones together for the record and Caveman suggesting that he tried the Jazz to get a rougher, bassier rock tone that he felt he wasn't getting from the Wall. Regarding the influence of Bruford and that music scene, I know that Rush were well aware of it. Jeff Berlin remains a good friend of the band and Geddy said in interview that Jeff was "the best bassist of all time", or words to that effect. Neil Peart even played on Jeff Berlin's 1984 album 'Champion', on the cover of the Cannonball Adderley song 'Marabi'. However, anecdotal evidence suggests the session didn't go all that well and Steve Smith ended up drumming on that song too. It seems in the end that both drum tracks were mixed in.