No from the language of agreement you just need to serve 30 day notice and for rest 30 days you can pay or adjust salary so in such situation company is wrong in asking you to stay later then 16 jan if you are ready to adjust your salary.
Serve a legal notice along reminder stating the clause and reminding of resignation letter and seeking full and final pay and the experience letter the company shall take do needful donot worry otherwise compensation can be claimed from them.