Assuming the single template argument version. If T has an implicit constructor taking type R as argument, wouldn't the second argument into max of type R be implicitly converted to T before getting passed to the max function? So there is no need for a special version taking both L and R?

This is a property of c++ implicit constructor rules, it is not unique to this function. In most cases this is something you want to avoid but for integer promotion it can some times be useful.