The first type of strength training is max strength. It’s normal not to feel any muscle soreness because the increase in strength comes primarily from fiber recruitment.
I would call the second type muscular endurance training or “power endurance”. The latter label is more common among climbers. Pure endurance training would be much easier.
The second type of strength training is definitely useful, especially for climbers. When to include it depends on how much easier training you’ve done to support it. As I understand it, typically power endurance training would happen at the end of a program after a base of both easy endurance and max strength has been completed. Including it too soon will lessen the gains that you get from it.