Vanity publishers are not terribly keen on admitting what they are. They masquerade as commercial publishers and it can be difficult to identify them for what they really are. Here's what you have to remember.
Commercial publishers make their money by selling books to readers, while vanity publishers make their money selling vanity publishing to writers. So, in order to distinguish between the two, all you have to do is work out who the publisher is selling to, and who they earn the bulk of their income from.
Commercial publishers like Random House and PanMacmillan use their websites to promote and sell their books to readers. They make it very difficult to find out how to submit to them; sometimes they even refuse to accept any submissions at all. They spend their advertising budgets on promoting the books that they publish. Consequently, they earn their money by selling books to the book-buying public.
Vanity publishers, on the other hand, aim their websites directly at writers, and actively solicit submissions. They accept submissions across all genres; they often comment on how difficult it can be for new writers to find a publisher willing to take them on. They spend their advertising budgets trying to attract new writers. Consequently, they earn their money selling vanity publishing to writers.
Vanity publishers will of course insist that they are not vanity publishers. They might call themselves subsidy, joint-venture, shared-responsibility or cooperative publishers, but from now on, you will not believe them. So long as they make the majority of their income from their writers, and not from readers, then they are vanity publishers, whatever else they might call themselves.