Then I mowed her lawn yesterday. When she paid me $$30$ again, I reminded her that we had agreed to $$40$ per week. She said that we agreed to $$30?!$

You're right, she owes you $$40$ per week. You were the offeror and you set the terms. By signing, she was obligated to pay you the full amount.

Ms$.$ Black is right. When she changed the price on the contract, that was her counteroffer. She terminated your offer and became the offeror.