“Am” is a linking verb and linking verbs should be modified by adjectives such as “good.” Because “well” can also act as an adjective, it's also fine to answer "I'm well," ...
Not quite. If you translate "am" as "soy" in Spanish, you can link it to "bueno" or "good".
How are you at writing? I am good.
If you translate "am" as "estoy" (like answering "como estas") in Spanish, you link it to "bien" or "well" when grammatical analysis tends to an adverb of circumstance, like generally well.
How are you? I am well, thanks.
If grammatical analysis tends to adjective of non-stable quality, on the other hand, as when asking someone who was previously ill or hurt, you do answer "bueno", I think, in Spanish, but idiomatic English is not "good" but "fine".
How are you? I am well, thanks!
How are you (now)? I am fine thanks!