What does it mean? Does it mean that implied volatility does not change with changes in underlying?

Yes, i think you have the correct idea.Derman introduced three approximate rules for what happens with the implied volatility curve when the underlying changes. The sticky strike and sticky moneyness/delta rule are the most well-known:- Sticky strike means that the implied volatility for every STRIKE remains the same; so the implied volatility of every option stays the same. This can be visualized as a fixed implied volatility curve when plotted as a function of strike- Sticky moneyness (technically, sticky moneyness and sticky delta are sligthly different!) means that the implied volaitlity for every MONEYNESS stays the same. Therefore, the implied volatilities of fixed strike change. In fact, this change can be visualized as a horizontal shift of the implied volatility curve in the direction of underlying, when the implied volatility is plotted as function of strike. Obviously, when implied volatility is plotted as function of moneyness, it will stay fixed.