In Type 1, it is certainly a battle challenge, in both cases. It can change to a rescue attempt during (if you negate the soul or gain access, for example), but it starts as a battle challenge and, barring a change as mentioned, would end as one for the purpose of triggers and effects that rely on "rescue attempt".
In T2, it gets a little more wonky due to the rescuer's choice, but it would be a battle challenge or rescue attempt based on your access to the chosen soul.
