Surveyor, Any dictionary that uses the word “causal” to describe correlation should not be trusted. I would go more with the statistics definition you had above. To take example of smoking and lung cancer…..There is a very high positive correlation between smoking and cancer but nowhere has it been proven that smoking causes lung cancer. The opposite: where non-smoking does not cause lung cancer is not true either.

To go back to the assertion from PBS, there might be a positive correlation between homeowner and their children getting better education. But that does not mean the buying a home causes your children to become smarter or get a better education.
The actual cause might be different, maybe folks who own homes are the ones with stable families and tend to spend time with their children. Maybe deadbeats, drug dealers, gang folks are not really interested in buying a home.

The fallacy of implying causation from just correlation was used in public policy with disastrous consequences. PBS is a serial culprit in this regard.