And, I guess, that you believe that there’s a scientific establishment that adheres to that method, and that the results it generates are disseminated faithfully to society at large.

I (mostly) agree with your definition, but the word 'ultimately' does a lot of work there. Many people who use 'believe in (the) science' are actually asserting that science already revealed the objective truths of the matter in question.

Why do you need to have faith ? The scientific method is all about proof.

At some point, you just have to believe, if you want to go much deeper, you get into epistemology. https://en.wikipedia.org/wiki/Epistemology

That is precisely the point where Science ends. The notion that Science can and should provide a full account of the Universe in all its complex glory is a very strange idea, conveniently promoted by those holding the credential purse strings, aka the Science hierarchy.

Proof is changing over time.