There is a real problem where as humans we are really only looking for 'alien intelligence' that we can interact with.

We would have a hard time identifying a solar plasma based entity that had reaction times in years.

But redefining 'intelligence' to include all complex processes is not useful. I don't care if a storm in a gas giant is just as complex as a chimpanzee's thought processes.

A rabbit is intelligent compared to an ant, but we wouldn't consider discovering aliens at that level of intelligence relevant to us.

What we are looking for is near-peer intelligent life. Meaning about as intelligent as humans at timescales competitive to human cognition.