It should be from 19 Dec 2013 to 19 Fev 2014.

You can refer to the example in the link below

For example, Principal Card approved on 15/11/12.
 
1st  quarter  = 15-11-12  until  14-02-13  (3 months).
2nd  quarter  = 15-02-12  until  14-05-13  (3 months).
3rd  quarter  = 15-05-13  until  14-08-13  (3 months).
4th  quarter  = 15-08-13  until  14-11-13  (3 months).

http://www.aeonmalaysia.com.my/acs/af/kona...814#page=page-1

So, your Card Dated 17 Dec 2013 (same as mine) was approved today after 2-days is not too bad. No idea how long it would take the Card to be couriered to Sarawak, probably a week or so from today. Lets hope u get is before Christmas or latest by the New Year.
There is no 'GST Waiver' for the AEON Watami CC. No special current promotion.  This card is good for the first RM1,666 spending a month of RM5,000 per quatrer where u get 3% CB (Min spending is RM3K per quarter to qualify)

you still get the 3% cash back if your quarterly spending is at least RM3,000 and capped at RM5,000 per quatrer.

3% returns every quarter is still better than FD returns.