How important are headings in SEO

Give google a slightly altered page that contains the H2 tags. But I think this falls into the "Black Hat SEO" category. And I don't want that.

You are correct, that is considered black hat by Google. They call it "cloaking".

Put the heading. Hide it with CSS. Everyone wins, including accessibility tools witch can determine the structure of the page using the "invisible" heading. But I do not know if this is considered a good practice.

That is also considered black hat by Google. They call it "hidden text".

Forget about the heading, and find other ways to tell google that we really are about posters. (via keywords, but do they really have the same "weight" as an h2 ? )

Meta keywords carry no weight at all with Google. You should be putting important keywords into the <title> tag. Google values text in the title tag more than anywhere else on the page, including headings.

Adapt the design to our SEO need, and find a way to put that heading nicely.

Before you do that, let me say that the need for heading tags is much less than it was three years ago. Three years ago the formula for ranking for a keyword was to build a page about it with the keyword in the title, headings, and used several times on the page. You would then link internally to the page heavily to boost the page's pagerank. You would always use the keyword as the anchor text. You would also build external links to the page using the keyword as anchor text.

Since Google has rolled out the "Panda" algorithm, using that formula is a recipe for disaster. Now to rank for a keyword you need to create several pages about it covering several aspects to the keyword on each page. You need to use title tags that include the keyword, but also include other text about the keyword. You need to use the keyword naturally a few times on the page. If you go overboard with the keyword, especially in headings, Google will actually rank you lower because you are keyword stuffing. When you link to the page, you should be very careful not to use just the keywords as anchor text. Especially on external links.

In short, Google is paying much less attention to on-page factor than it used to, and penalizing much more often for keyword stuffing and getting spammy links. I'd go so far as to say that heading tags are optional now. They may help some when used in moderation, but they are much more likely to hurt when you put your keywords in them.

2019 Edit: At this point heading tags are even less important than they were when this answer was written in in 2013. Google now renders pages and it will give more weight to any prominent, large, bold text at the top of the page. It doesn't matter whether or not that text is marked up with heading tags at all. For SEO, heading tags just don't matter anymore.

Using heading tags can still make your page better accessible to blind users with screen readers. Screen readers usually have functionality to read headings first and jump to the specific section of the page under a heading. Using heading to give structure to your page is important to visually impaired users. Use headings for users, not for search engines.