1. You need to give only 7 days notice as stated in appointmentt letter
2. 7 days is sufficient, it means seven working days and does not include weekends although it is not defined so anywhere in statute. The day you tendered resignation is not included.
3. You can use balance annual leave entitlement in proportion for the part of the year so far to offset the shortage in notice period.
4. HR is likely to force you to serve 2 month notice but you need to stress the fact that there is no other communication addressed to you either confirming you in service or extending your notice period to two months. Unless these are specifically communicated, no presumption and assumption could lie. You need to softly challenge the HR Department that legally there cannot be any claim for two month's notice and you will not be cowed down. HR may reject your resignation in that case, issue them proper legal notice.
21st June 2018 From India, Mumbai