I agree that the first example in #1 isn't fraud, though it could be illegal for other reasons.
'Fraud' is a legal term as defined in the laws of a particular country or state, in a fairly formal register. You can be arrested for fraud or convicted of fraud. It can be either a noncount or count noun: the general concept of deception for gain or a particular instance or type of it. The related verb is 'to defraud (so. of sth.)'; the adjective is 'fraudulent.' More casually, you can say that a person is a fraud, that is, a fake.
A scam is a slang term, typically used in conversation but not in law. It may refer to the same thing, just in a different context or register. It usually means a particular trick, e.g., the Chinese college scam, the Nigerian loan scam, the lost-grandchild scam. 'To scam (so. out of sth.)' is also a verb, also colloquial. A person can be a scammer or a scam artist, similar to a con man or a con artist.