It was my impression that adding a new specialization made the 4 bonus career skills from the specialization count as career skills for you character. In talking with another player, he thinks it adds the skills from the specialization's parent career to your list of career skills as well. Looking at the actual rules under 'Acquiring New Specializations' on pg 93, it says

Purchasing a specialization is basically the character buying the ability to purchase talents within that specialization. In addition, each specialization has career skills. These skills now count as career skills for the character (although he does not gain free advances in them, as he did with his first specialization). Characters may purchase any specialization in any career.

Now if you look at any of the talent tree pages, they're all of this format:

<Career name>: <Specialization> Talent Tree

Career Skills : <Stuff>

<Specialization> Bonus Career Skills : <More stuff>

It strikes me that if adding a specialization to your character just gets you the 4 skills specifically associated with that specialization, the underlined sentence in the passage I quoted should say 'In addition, each specialization has bonus career skills. These skills now count as career skills for the character (...).' Yet the passage just says career skills and I'm not finding a specific rule that says which interpretation is correct.

So what's the answer? Is there an official example of adding a specialization from outside a character's career that shows what the character's subsequent skill list looks like?

the 4 bonus ones.

Agreed, just four bonus ones.

It was my impression that adding a new specialization made the 4 bonus career skills from the specialization count as career skills for you character. In talking with another player, he thinks it adds the skills from the specialization's parent career to your list of career skills as well.

The other player isn't making much sense: if you buy a second specialization within your career you only get the 4 bonus ones, since you already have the base skills. An extra 10XP for an out-of-career specialization wouldn't begin to cover the cost-benefit (every rank being 5XP cheaper) of getting all the base skills too. In fact, if the other player was right, it would make zero sense to ever buy another in-career specialization.

You can't buy another career -- your starting career is it. Ergo, only the skills associated with the new specialization count as career skills for your character.

Agreed. One of my group recently bought a Doctor Spec for his Outlaw Tech. Just adding the four career spec skills made a thick black column of his career skills list. I can only imagine how unbalanced that would make the XP expenditure system in the long run...

Side note, though I think you already noted this, you don't automatically get a free rank in two of your new career skills; that's only at character creation when you choose your starting career and specialization. So that helps balance it further.