Again, you are being pedantic. Dixon didn't want to work for Barnes, and for all the evidence shown thus far, Barnes wasn't wild about having Dixon work for him. Even if the worst description warranted is that they were completely neutral toward each other, holding no ill will whatsoever, Barnes wasn't going to fight to keep the coach.
It doesn't matter which person in the couple said, "we should see other people," first. The relationship was already over long before that happened.
It's nothing but a chicken and egg scenario to say that Dixon was unhappy with lack of support from Barnes, or Barnes was unwilling to show unrestricted to support to a coach who was seemingly declining. Barnes was hedging his bets that boosters were unhappy and he might have to fire a program-changing coach in a couple years. Dixon wasn't willing to risk his career by coaching under an AD he didn't like, for boosters that didn't want him.