So, back to Rabbit 1.

[05.015] xx and when he said gollum xx he made a horrible swallowing noise xx in his throat. xx That is how he got his name,

Is that how he got his name from Bilbo?? Or was that something that the goblins called him? And in the latter case, then how could Balin say:

[08.127] xx ‘Gollum! Well I’m blest! 

Hmmm. I think that it must be Bilbo who created the name when he told the story of his escape from Chapter Five.