Rodgers did not invite Favre according to his version of events. The NFL invited him to the banquet and asked Rodgers if he would present an award with him. They presumably asked Favre the same thing.
Murphy, minus the acknowledgement of Rodgers/Favre joint appearance, has been making the same gesture to Favre for 3 years running, if not longer. In fact, he makes is annually during the tailgate tour.
And Favre is not a reliable narrator in this instance. He formerly said he didn't harbor a grudge and it was just business between he and the Packer. Admitting a grudge existed might be evidence of his being more truthful publicly, but its certainly not determinative.