this list is not comprehensive.
From what I can find online, there's a correlation, but it's pretty light on details about how strong it is.
> Sense of "indisposition involving catarrhal inflammation of the mucous membranes of the nose or throat" is from 1530s, so called because the symptoms resemble those of exposure to cold; compare cold (n.) in earlier senses "indisposition or disease caused by excessive exposure to cold" (early 14c.), "chills of intermittent fever" (late 14c.).
Put less technical: shivering, runny nose, etc, are symptoms of both the common cold and being cold.
It seems it's best to think of "cold" and "flu" as totally informal terms with a wide range of possible meanings. I've had Actual Real Influenza twice and both times it knocked me on my ass for a week, lost weight, etc. yet I hear people saying "I have the flu" when they have mild symptoms and only miss a single day of work. Hard to take such a wide range medically seriously.
They could have had influezna with mild symptoms. Two different people can have the same influenza virus enter their body and experience different effects. Anywhere from no symptoms at all to death.