Dear Sinmranjeet,
Specific to your situation, here are the answers.
1. Correct, Sec 257 is not applicable to a pvt company
2. Correct, in case of a pvt co you can appoint a director without going through the Sec 260 route (additional director), provided your articles permit.
In this regard, Sec 255(2) is relevant.. to quote,
The remaining directors in the case of any such company, and the directors
generally in the case of a private company which is not a subsidiary of a public
company, shall, in default of and subject to any regulations in the articles of
the company, also be appointed by the company in general meeting [emphasis supplied]
The import of this provision, as related to private companies is that every director must be appointed by shareholders, in default of and subject to provisions in the AOA. Therefore, if the AOA provides for direct appointment, then you can do so.
3. In your case, there is no benefit of this provision, because the AoA says such director will hold office till following AGM. And the reference in the AoA is clearly to 'Additional Director' .
4. Suppose your AoA had mentioned that 'The Board, from time to time, shall have power to appoint one or more persons as Directors of the Company to hold office for life and the provisions of Sec 255(2) shall accordingly, not apply' or something on similar lines, then, yes, you can appoint a person directly as a Director without going through the additional director route. Needless to mention, as you have rightly pointed out, Sec 257 is inapplicable to a private company, and hence, there is no further need of shareholders approval.
Please also note that Sec 161(1) of the Companies Act 2013 relating to additional directors has already been notified. Sec 257 of the 1956 Act still continues as clause 160 is yet to be notified.
Hope this clears your doubt
Vinod