Zanzibar was not a “third country.” The whole region was British-controlled but with differing structures. Mombasa was “nominally” under the control of Zanzibar, which, I believe, had at least the notion of some “autonomy” rather than a direct British colony. But Zanzibar was under heavy British influence. Eventually most of Zanzibar went with Tangyanika to form Tanzania but Mombasa went to Kenya. But it had long-standing ties to Kenya. In 1961, going to Mombasa from Kenya was not exactly “going to a third country.”
Yes, the whole region was British-controlled but at the time Obama was born Mombasa was not technically part of Kenya. The whole idea of Obama’s parents leaving Hawaii while she was pregnant never made much sense to me anyway. But it only made sense to me if I could believe that maybe Obama Sr. was bound and determined that his son be born in his home country of Kenya. That’s the only explanation I could think of for making such a risky trip halfway around the world with a young, pregnant wife. So why go through all that and then have the baby somewhere that was NOT technically part of Kenya?