I think there are two questions here.

First, this abuse of terminology regrading a) the volatility term in the equation describing the dynamics of the process, $dS=rSdt+\sigma S dW$, which is sometimes referred to as the instantaneous volatility, and b) the volatility of the price itself is quite 'standard'! Probably because in most cases the meaning is clear from the context, though that's not a great excuse. Also you will hear of the total instantaneous volatility, which is $\sigma \sqrt{T}$.

A related confusion arises in the terminology around the diffusion coefficient - finance people would identify the diffusion coefficient as the coefficient of the brownian in the SDE, say $\sigma$, whereas physicists would would identify it as $\frac{1}{2}\sigma^2$ (think the diffusion equation).

Second, as @will commented, the precise statement regrading the breakeven is in the approximation sense. As a simple explanation, recall the Brenner Subrahmanyam's approximation of Black Scholes:

$C*0 \approx 0.4 S*0 \sigma \sqrt{T}$

We can approximate the price of the straddle by just doubling the call price:

$\mathrm{Straddle} \approx 0.8 S_0 \sigma \sqrt{T}$

So to recover the cost, the price has to move by roughly $\sigma$ percent.