Bamboo grows wherever you put it, we have two large stands which we put in the garden eventually in very poor soil - it's eight feet high now. JI is good all-purpose compost, but bamboo's not fussy.
The only reason they may of suggested JI3 was because it's a loam based compost as opposed to peat based and is therefore heavier....something that needs to be thought about when growing top heavy plants in pots.
John Innes refers to a formula ... As in JI potting, Nos1 ,2 and3 and not a specific brand
The amounts of "food" required by plants changes as they mature, and each type contains differing amounts of NPK ... Originally formulated by the JI institute
As I usually comment , if you want to know , then access the RHS site and ask the experts!