Correct. In my ice-cream example, the third variable is obvious. (Right?) It's a perfect illustration of why correlations have absolutely nothing to do with causality.
Actually, "ice cream and crime" is a famous thought experiment which conclusively demonstrates that, in certain cases, correlation
can prove causation. They teach it in stats classes.
Short version: Ice cream sales and violent crime are correlated. This could mean that ice cream sales cause more crime occur, that crime increases ice cream sales, or that a hidden third variable causes both ice cream sales and crime to increase.
Well, if you were to look at some other variables, you might also notice that a decrease in the heroin supply is also correlated with an increase in crime; but interestingly, when crime rates increase due to changes in the heroin supply, ice cream sales are unaffected. That would effectively disprove the notion that crime causes ice cream sales. Similarly, the fact that birthday parties for six-year-olds are correlated with increased ice cream sales but not increased crime would contradict the claim that ice cream causes crime.
You might also notice a correlation between temperature and both ice cream sales and crime: on hot days, there are more violent crimes and more ice cream sales. Rainfall is also correlated with heat, and interestingly, when there are changes in heat due to rainfall, there are corresponding changes in both crime and ice cream sales; but when there are changes in ice cream sales or crime due to birthdays or heroin, there are no corresponding changes in heat or rainfall. This could be considered scientific proof (but not mathematical proof)* that heat is causally related to both crime and ice cream sales.
*It's semantics, but it's an important distinction.