As you said, I may required to provide evidence for my gap year. But how could I provide evidence without mentioning about Cambridge? Because the fact is that I will have to stay in Cambridge for one year. So, no matter I take language courses or work in an institution, it will be in Cambridge, and only Cambridge certificate can proof that I will stay there for one year...I do not think that you are required to show evidence for another programme. You can state, in the letter, that you are applying for some positions that would give you practical experience and you would not hear back until after the deadline of responding to this offer. So, you decided to defer your entry to next year; rather than rejecting the offer.
They may even think that you intend to live in Japan for one year before joining the programme. Do not write this, allow them to think anything.
I think that they may understand that you want to be on the programme, but you need some more time.
Did Columbia ask for evidence to support your application for deferral?? If they did not, then it would be fine to just state it without mentioning Cambridge.
And in this deferral letter, should I just generally state that "I wish to gain more academic experience in order to be fully prepared for Columbia program", or shall I write in detail, including what is my research interest and what I am going to do (take language courses) in the year I gap.
I think that you should state that you wish to gain some practical experience (this can be academic or otherwise). If you write academic, they may think that you chose to study at another university, which I think would not be good.
I think that you need to write, in detail, what you hope to develop during the gap year and how it would help your Columbia program.
For example, by taking the gap year, you hope to understand the different Japanese dialects and how they influence effective communication. This knowledge would be very beneficial to my research at Columbia because I believe that I would have acquired some knowledge on interaction between the Japanese and other cultures.
Do you think I should write to explain my case and ask if it is possible for a deferral (which means I am still thinking about it and I want to know if the admission committee allows me such a gap year. -- Therefore the decision is not confirmed. If they say no, I still have a chance to take the course this year. But there is a disadvantage. They would probably think that you haven't decided yet therefore they would probably say "no" in order to force you quit your current job or your gap year plan.)
Or I should wrote "I have decided to gap year and I am writing to ask for a deferral." For this, if they says no, then it means that I lose the chance to get into Columbia this year?I think that this would be tricky. This is because if you ask, as you wrote, they may reject you all together. On the other hand, if you apply for the deferral they may also reject you. I do not think that there are guarantees. You would need to make the decision. Since you have decided to take the Cambridge place, you should be ready to lose the Columbia place (worst case scenario).
If it was me, I would apply for it and see what happens.
If they approve the deferral, then you are lucky. However, if they do not approve it and, may be, reject you all together, then you can apply next year. Besides, I think that the Cambridge name could help the application.
Let me know how it goes.
Good luck