How to use I or Me it depends on you, but if you are writing active sentence mostly you use "I" and if you are using passive sentence then you use "Me"
"I" am playing football. Active Sentence
Football is playing by "Me". Passive Sentence
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I concur with Paul. Either "I am" or "Me" works fine in my SE Pennsylvania English. If speaking more formally and politely, I'd probably answer "I am", and if speaking more informally and casually, I might be more likely to say "Me", but they are both equally good.
The same thing works in other cases of wanting to respond to other questions with a short first person pronoun answer:
Who wants to go? I do. / Me.
Who's hungry? I am. / Me.
Who did this? I did. / Me.
Who likes ice cream? I do. / Me.
Who has $5? I do. / Me.
And the negative answers:
I'm not. / Not me.
I do. / Not me.
I didn't. / Not me.
And the same choices work for other personal pronouns:
We are. / Us.
They are. / Them
He is. / Him
She is. / Her
And maybe you're right. Maybe "Me", "Him", and "Us" are subconsciously short for "It's me", "It's him", and "It's us".