The terms unconditional life and unconditional death are somewhat ambiguous and can mean different things depending on context.
In the example above, the first solution is considered a failure, because it does not kill unconditionally. The outcome depends on a ko. The second solution is considered correct, as black dies unconditionally.
In research articles, unconditional life sometimes means alive even if the defender continues to pass. This is opposed to alive, unless the defender continues to pass. Here the condition is alternating play. This usage of unconditional life can be helpful for beginners to understand the concept of alive. It is also the common usage in more scientific approaches of life and death issues in game positions, e.g., in articles that lay a base for computer algorithms such as Benson's algorithm, but it can be used in ordinary game discussions too.
This meaning is also referred to as Pass-alive, absolute life or invulnerable. As beginners have been known to fill in the eyes of their groups, there is practical reason to prefer the first term.
More at /Discussion.