word choice - How exactly does one determine when to use I or Me?

I got into a good argument with myself when a Lecturer asked:

"Who said that?"

and I replied

"I."

Actually, I didn't just want to stop there, but I felt there was no need to continue and that it was correct. However some argued that "Me" was the correct reply.

Since the reply was not a sentence, it was difficult for me to conclude whether I should refer to an objective Me or a subjective I.

So in short, I want to know whether there is any other clear parameters that defines when to use which and how?

Answer

Either works fine in this case, although me is better. The person who responds with I is actually saying I did but holding back the did. The person who responds with me is simply using the customary emphatic form.

But yes, certainly I is for subject forms and me for object forms. There are just a few situations where what appears to be an object form (but isn’t), are called for, such as “Me, I wouldn’t say it works that way.”

Note that French works the same way in this regard, whereas Spanish does not.