To save you loads of discussions on which of the two–AmE/BrE --is ‘more correct’, just zip through this: … 20.article

It’s really informative and interesting. But as regards the following (in particular), I am not sure of the usages in bold.

In British English, the auxiliary do is often used as a substitute for a verb when replying to a question, e.g.:
A: Are you coming with us?
B: I might do. (Is this really the way it is answered?)

In American English, do is not used in this way, e.g.:
A: Are you coming with us?
B: I might.

In British English needn’t is often used instead of don’t need to, e.g.:
They needn’t come to school today.
They don’t need to come to school today.

In American English needn’t is very unusual and the usual form is don’t need to, i.e.:
They don’t need to come to school today. (Doesn’t it give a slightly different meaning?)

Sometimes, yes.

I don’t see a distinction.

But I see.

You use Indian English.
There is no distinction in British English as far as I am concerned. They are interchangeable.

Why ask if you have already decided?
Why not just explain what you think the distinction is?

Yes, as far as I am concerned, I watch the trends in InE, BrE and AmE. I analyse them and accept what is convincing to people of ordinary prudence. I’ll also try to share with others what I observe and effort to get it recognized if possible. You need’t worry about what I don’t need to do. LUSH.

Oh, dear me.
Such delusion.