This site uses the word oscillator to represent a component with an self-contained oscillating feature that has power, ground, and signal out pins. This site uses the word crystal to represent an component which contains a oscillating element (in the form of a crystal), which requires an external oscillation circuit before it useful.
A common designator prefix to use for oscillators is \(Y\) (e.g. \(Y1\)). I do not recommend using the prefix \(XC\) as this should be reserved for crystal oscillators.
Phase noise is a way of describing the stability of the crystal in the frequency domain.
The start-up time for most oscillators is within the range 2-20ms. This start-up time can be important in low-power designs when the start/stop time of the crystal results in wasted energy.
The smaller SMD oscillators use MEMs technology in order to fit the entire device into small component packages.