I define “science” in the second way.
Often scientific authorities will tend to define “science” in such a way as to not allow supernatural causes. This is normally good, because science is a process or method of using natural means to measure natural things or events. The method of science can be used to come up with the most reasonable explanation for things or events and thus naturalistic conclusions can be understood, but it cannot completely rule out any possibility of a supernatural cause. We want to always keep searching for the most reasonable cause and sometimes a supernatural cause is much more likely based on the conditions and probability.