The restored capacity of a battery
Could anyone please explain what the 'restored capacity' is?
  Is it different from the capacity of a battery at certain temperature and certain (constant) discharge rate?


No. Probably.

It is very likely a matter of "Lost in translation".
I'd expect that words like "achieved" / "returned" / "actually provided" / "real-world-results" / ...  - are what they are saying.
ie the cell is nominally 5Ah at stated conditions
(25 hour discharge rate, 20C ambient, 2V end of discharge point) BUT if you discharge it faster (or slower) and/or at a different temperature or using a different end point voltage then Your AmpHourage WILL Vary. 

The bottom graph on page 2 shows that at eg 20C you get
5Ah at 100 mA discharge rate,
~= 4.9 Ah at 500 mA,
~=4.6Ah at 1A, &amp;
~=4.2 Ah at 2A.   

These are the restored/achieved/predicted/... capacities.