How to say lived vs. have been living somewhere?
I would say, as a Chinese, that Chinese language doesn't express tenses as explicitly and clearly as English does in most cases. Your translation of the first sentence(我在北京住了六个月) can be taken in both ways, in which case we figure out the tense by context. Past tense would be a better guess, though.

To remove uncertainties, you can say:

我在北京住过六个月。I lived in Beijing for six months.

我在北京住了六个月了。I have been living in Beijing for six months.

Or for added clarity:

我曾在北京住过六个月。I once lived in Beijing for six months.

我已经在北京住了六个月了。I have been living in Beijing for six months already.