"believe in science" What does this even _mean_?

It means think what someone else wants you to and don't question it.

Believe that the things that science says are true actually are true. It's pretty clear, no?

Is light a wave or a particle?

It's something that has wave-like and particle-like properties. What's your point?

That you have faith in the ability of the scientific method to ultimately reveal objective truths about the universe.

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.

You dont need to believe in anything. Science is not a dogma. if Science is correct it can be demonstrated.

Unlike heating up balloons filled with different gases - climate change isn't something that can be demonstrated easily. I don't disagree with your statement in general, but when it comes to climate change I'm offended by any contrarian who passively sits back in their chair and shouts "prove it" - you aren't owed that.

Why that person that you are offended by cannot be offended by your statement of active screaming (sorry, just use your terminology)?

We never did, but it can look as if it is moving further away from that goal