The main source we have for the Battle of Cannae is the account written by Polybius, a few decades after the battle actually took place. The other major source is that of Livy's, which was written even later, and perhaps drew on Polybius' account as a source.

Given how thorough the Roman defeat was (leaving relatively few living participants and eye-witnesses), and how much time had elapsed between the actual battle and when these accounts were written down, there is some scope to doubt the accuracy and veracity these accounts, especially when details do not seem to jive with/match with what's known to have been Roman practices.

On top of that, Polybius' impartiality is suspect, as he was in his youth a hostage in consul Paullus' household, and had a client-patron relationship with that family.

So, the accounts tell us that Paullus was commanding the right wing of the Roman army. But this was traditionally where the Roman commanding general would have placed himself, so it's inconsistent with Varro being in command that day. Further, while Paullus stayed and fought to the death, Varro fled and was received rather curiously by the Romans, who thanked him for his service and expressed gratitude that he didn't abandon Rome in her hour of need. This from a people who are not known for having much sympathy or patience for military ineptitude.

It's been suggested that this is consistent with Paullus having been the one who desired pitched battle, and therefore alternating commands, while Varro sought to continue the Fabian strategy of dogging Hannibal while declining battle, and so argued for a divided command arrangement. And that the Roman Senate overruled Varro and forced an alternating command arrangement on him (which, to be fair, I believe was the traditional/usual arrangement). Paullus stayed and fought to death because he took personal responsibility for the disastrous defeat he'd led his army into. And the Roman Senate thanked Varro because they realized he'd been right all along, and they erred in forcing that command arrangement.

This is all speculative/alternative theory though. To my knowledge there is no direct, primary source evidence that says Paullus was in charge day. Just that some details seem to fit the alternate narrative better. And the sources we have can be said to be a bit suspect.

Thank you for taking the time to answer. Interesting stuff. One thing though. Can we really assume that Livy and especially Polybius could be wrong/lying on a subject? Polybius was almost contemporary to the event. One might think that at a time he wrote his Histories everyone still knew quite definitely who was in charge.