What does 'random phase' means in a triac driver?
Yes, it means it doesn't have zero-crossing detection. The latter is useful for purely resistive loads, where the current is in phase with the voltage. For reactive loads, inductive or capacitive, this doesn't offer an advantage.
"Random" just means the triac switches on at the time you signal it to switch on, whatever the voltage phase at that moment. So its actual meaning is that it can switch on at any time.  


  A note I made earlier on zero-crossing switching:
  You may have noticed that incandescent bulbs always fail when they're switched on. That's because the mains phase can be near its maximum when switching on. Combined with the low resistance of a cold bulb this results in a high current peak, which may burn the filament. When you switch on a zero crossing you avoid these peaks.  


Random switching is needed if you want to build a triac-controlled dimmer, where you want control of the phase angle where you ignite the triac over the full 180° of a half cycle.