Tikk - speaking of "exclusive rights", check out the deal cut by the lawyer for an ABA franchise about to go under pre-NBA merger. Google Donald Schupak and the St Louis Spirits. The story is worth 10 minutes of your timeThe answers to many of these questions would require looking at the contracts themselves and that's not going to happen here.
The term "exclusive rights," to borrow from the "bundle of sticks" analogy, might not necessarily mean all the sticks, it might merely mean the right to broadcast live. It can have legal import in the context of a contract, but we hear it more in a marketing context; when I hear the term I don't typically presume much by it. But I wouldn't imagine that Company A would be using the term if Company B has rights to simultaneously broadcast on a separate distribution channel.
I too am curious what BATS-TOI (thought they) paid for. Also curious why they thought it was a good idea to call their company/product BATS-TOI; reads like an acronym for baseball statistic I couldn't possibly care about.
The moral of that story - Never give away the rights to something with an unknown future value.