In theory, yes. But when you include hedging costs and taxes, the link becomes less direct. The cleanest way to get long volatility is by purchasing ATM straddles.
With a put, you primarily pay for directionality with hedged upside risk: you don't lose your house if the stock moons. While it's true volatility is a component, that's a side effect of the hedging since your counter party takes on volatility risk.
Buying puts is more about longing volatility.