I take it there's more than one, as the logic book I am studying mentioned the 'material biconditional', and that the term material can be dropped for just biconditional in general.
So presumably, there are 'non material bi conditionals', or other types of bi conditionals that perhaps have different truth values? Or none at all?
After some research it doesn't seem that there's any other type of bi conditional, apart from the one stated in sentential logic (which of course carries into other or/and more advanced logic, like predicate logic).
Wikipedia tells you about material conditional, indicative conditional and counterfactual conditional.