> the term used when doctors find a gene that runs in the family and causes cancer
Gene's can be switched on an off and gene are by nature inheritable. Not all genes mind otherwise we would be identical.
Lynch's syndrome is not a general term but a specific inheritable genetic mutation that affects about 1 in 270. It puts them at higher risk for certain kinds of cancer. The reason to get tested, if you have one or more relatives who have suffered from colon cancer, is to do early and ongoing screening for colon cancer. Lynch is more common than the BRCA3 mentioned in the article.