First and foremost let me comment on the choice of your words. You have used the word indulged. Do companies engage in manufacturing activities or indulge in it? I request you to check meaning of the words before their usage. Choice of words is very important gentleman/gentlewoman. Else it could land you in trouble!
You have two options. One offer is confirmed and it is in private sector. The other is not offer as such. It is your desire to pursue career in banking sector. In former you can rise little faster. On the flip side, the job guarantee is less. In contrast, in government banking sector, growth may be steady but job guarantee is there. Therefore, more than job I recommend you looking at the your career ambition. Above all, at this stage, you are only appearing for the competitive exams. What is the guarantee of success?
A bird in the hand is worth two in bush goes the famous proverb. Whether to live it up or defy it, the choice is yours!
As far as questions are concerned, in addition to 2-year bond period, you need to ask questions regarding the job profile, whom this position reports to, job challenges, Key Result Areas (KRAs) and so on.
All the best!