I think you're taking the wording too strictly. Beta is a measure of market correlation and when they're talking about volatility in that sense then they are talking about market volatility. That is to say that with no external changes with the company, if a stock's beta is 0.85 ad the market goes up 1% then the expectation is that the stock will go up 0.85%.

It's not suggesting that there is zero market independent volatility, it's simply not addressing it here. When you add in non-market-correlated volatility to any mid (or higher) beta stock then it will almost always be higher volatility than the index -- which is why people buy indices.