What exactly is primary rated voltage of transformer?
For example, we have a 2200/220 V non-ideal transformer.


OK. 


This transformer is designed to take 2,200 V on primary side. 
It is designed to give out 220 V on the secondary side when the primary is excited by 2,200 V.
Therefore it is a 10:1 transformer.



  Is primary rated voltage defined as the supply voltage to get 220V across the load in this case? Or it is just the 2200 V as mentioned in rating?


The primary rated voltage is that at which the transformer can pass maximum power.


If the primary voltage is reduced then you will no longer be able to get the same VA out of the secondary. Power output will be reduced. Remember that the windings will have a certain current rating so you can't draw higher current from the secondary just because you have reduced the voltage.
If primary rated voltage is exceeded the transformer will saturate and overheat.



  More generally, is primary rated voltage defined as the supply voltage we need to have so that the secondary rated voltage is appeared across the load?


Answered above, but yes, you are correct.


  Or is it just equal to primary induced emf rms value?


This doesn't make sense. Induced where and by what?