I agree volatility of the stock price is represented by standard deviation. When I day trade, risk and volatility are essentially the same. But as a long term investor, my #1 risk is the probability of the company going out of business. Of course there are also other risk, like I may not get all of my money back... It is often very complicated, different from just the volatility of the stock price? Anyway, it is beyond my pay grade, my capability, to define risk.
It's more nuanced. Risk is commonly defined as the possibility of something bad happening. And since this "something bad" is unknown, we tend to use volatility as an alias for it. When you're investing, longer your investment horizon (holding period), more can you treat daily, or even weekly volatilities as noise. And when you invest for the long term, you should simply put your money and forget that it's there (provided you've run due dilignce on the long term growth prospects of the investment). This way, any drawdowns won't make you divest and you''ll enjoy your LTCG. Now coming to trading, the major aspect of risk management is to have proper stops/trail stops in place. But since every stock has periods of high and low volatilities, you should adjust your stops accordingly, and not just have them at fixed positions. The essence of risk management is to ensure zero or minimal loss of capital. And when the above mentioned things are in place, your risk is mitigated. Volatility is just an attempt of quantifying risk. There are other methods as wel, for example, the likelihood and severity matrix, the VaR, and many others. Then there's the Sortino which takes into account ony the negative volatilities. Depending on your objectives (to create more wealth, to be risk averse, or to have a combination of both), you can choose to employ the metric that serves your investment or trading strategy best.
might a bit too simple to share but here is a good read in general: https://www.cannontrading.com/riskopportunity