I was wondering...

Bungie left Microsoft because they wanted to do something other than Halo. Maybe they already had Destiny in mind, told Microsoft they wanted to start a new franchise and leave Halo, Microsoft said no way and then Bungie decided to look for another publisher?

Would be really ironic if Microsoft refused to let Bungie make them a new IP which probably will end up being bigger than Halo.

Or is there any evidence on how this story really was?

I never quite understood Destiny's story. As far as I know, Bungie wanted to be independent after Halo 2, so then they signed a contract with Microsoft for three more Halo games and IP ownership (to Microsoft) and then they could do as they please.

After that they announced Destiny. Then it came those things about "Maybe Destiny is multiplatform!" or "Destiny will be on Xbox consoles a full year before releasing on others!" thing.

Then you have Destiny nowadays, which pretty much releases at the same time with other consoles, and has exclusive goodies on Playstation.