Cleared for the approach gives you the go ahead to start the descent and follow the profile.
"Cleared on XX 2X Arrival" to me (as a former en-route feeder controller) means that`s the approach you should plan for albeit phraseology is different than what I would give (You are cleared to THAT airport via the THIS approach).
Lackadaisical phraseology really annoys me, it is quicker to get it right the first time and leave no doubt in the pilot`s mind.
Last edited by LoserGill; 16th August 2012 at 18:59.