Good question! When you're not following 'I am' with anything it's best to say it in separate words or just to say 'yes' or 'no'. You can add 'I am' for clarity or emphasis. Are you X? Yes (I am) (In this case never say 'I'm' alone) Where are you from? I'm from X (In this case you can say 'I am from X' but it's more common to say 'I'm')

"Are you Ecuadorian?" "Yes, I am." (correct) "Yes I'm." (incorrect and unnatural... it sounds as if you didn't finish your reply.) "Yes, I am Ecuadorian." / "Yes, I'm Ecuadorian." (both correct)

The only one that is incorrect is "Yes I'm." because it is an exception to the rule. The same goes for : Yes he is. Yes she is. Yes it is. Yes they are. Yes we are. You cannot use a contraction for a short anwer such as these ones.

Yes "I am" and "I'm" are the exact same thing. You can use either one at any time.