There are plenty of stocks that have beta < 1 that are much more volatile than the indexes.
Having a low beta simply means its price isn't as correlated with the overall market. An example, Netflix has a beta < 1 but is extremely volatile.
Beta is volatility relative to the entire market. That's why I said "less volatile to market conditions than an index fund" Basically if the market has a big day, it will move up less than the whole market. Similarly on a down day. I think for both Netflix and Microsoft it's because they have strong brands and recurring revenue streams. They are also less dependent on external funding, so whether the market goes up or down has little impact on their P&L.
The converse, which you pointed out, is that they are highly impacted by non-market risks, which is very big, especially in the case of Netflix. There is a lot of discounting of future cashflows, so if something small happens (like splitting off the CDs business) then it creates a lot of volatility.
I think we're on the same page, no? Should I edit the original to be more precise?
Beta = cor(S,M)*vol(S)/vol(M)
Where cor(A,B) is correlation between A and B, and vol(X) is volatility of X (i.e. standard deviation).