I agree - how you personally define risk has no affect on market behavior. (How society in general views risk may affect stocks is a different story.)
How one defines risk does, though, change how one interprets the behavior of the market, though. If one defines risk as standard deviation, that affects how you interpret risk (vs. if you define risk as downside risk).
For example, if stocks suddenly went up 50% in a matter of months, the investor with the "risk=standard deviation" definition would see their portfolio's standard deviation go up. This person's models would tell him or her that their portfolio risk was skyrocketing. The investor using the "risk=downside risk" definition would not be told by their risk indicators that risk had increased. (Many feel this is a limitation of the risk=standard deviation interpretation- that it penalizes upside uncertainty).
Basically, definitions of risk affect interpretation of market performance - and thus definitions are important.