Is the Black-Scholes formula the only way "implied volatility" is calculated/defined in markets?
Unless otherwise stated, when people talk about implied volatility, they indeed mean the implied volatility under the Black-Scholes-Merton (BSM) model. In practice, it is quoted on a yearly basis and it's the information you would get from data sources such as RiskMetrics. It is also what people mean when they talk about estimating model parameters by (quasi) maximum likelihood using option prices: they are fitting the BSM-implied volatility surface.
I am not an expert on this, but I do recall an exception burried in the asset pricing literature. A very famous paper by Bakshi, Cao and Chen (1997) diagnose the capacity of models to improve on the BSM model by seeing if the volatilities they imply across moneyness and maturities is flatter (since you have just the one underlying, at some point in time, you have just the one volatility). It's an awkward way to put the problem which is probably why later papers that I have read seem to focus on matching the BSM smirk and not on finding a way to get a flatter one.
In other words, it's very unusual to run into other model implied volatilities -- at least as far as equity options are concerned.