The greeting How are you? is asking How are you doing in general?
—
How are you?
I'm well. [Misunderstood the question.]
because well as an adjective which means:
in good health especially after having suffered illness or injury
This would be an answer to How are you doing physically, how is your health?
—
How are you?
I'm good. [Misunderstood the question.]
because good as an adjective means:
having moral excellence or admirableness
This would be an answer to
How would you describe your character, superman?
I'm good.
—
How are you?
I'm doing good. [Grammatically incorrect.]
because good is an adjective, not an adverb.
—
How are you?
I'm doing well. [Correct.]
because well is an adverb describing how you are doing.
How are you?
I'm doing fine. [Correct.]
because fine is an adjective which means:
being satisfactory or in satisfactory condition
Why do so many people say one of the first three responses?
Best Answer
The reason people give the responses you label as “wrong” is because well and good have more meanings than the ones you cite.
Merriam-Webster, for example, gives definitions that make sense of the three examples you label “wrong”:
For “I’m well”, there is well (adjective) 1b: being in satisfactory condition or circumstances
For “I’m good”, there is good (adjective) 2e: free from infirmity or sorrow
For “I’m doing good”, there is good (adverb) 1: WELL
So they all seem fine to me.