Let me tell you the exact corporate practice following by top corporate and listed companies:
1. 1. For the appointment of simple director there is no need to execute any agreement. Simply collect a consent letter from them and file form 32 with ROC.
2. 2. For the appointment of MD or WTD etc some companies follow the practice mentioned in point no. 1 where terms of appointment are not very heavy.
3. 3. Where terms of appointment of MD or WTD including remuneration etc are heavy and exhaustive it is better to enter an agreement with MD or WTD.
You may also issue an appointment letter to director though it is not mandatory and dependent on the practice of the company.