I'm happy to agree to disagree about where the line between "division" and "division like" should be placed. As you say, it is a question of convention and not really a question of math. But I don't agree that a student with the curiosity to ask about extending the numbers in various ways would not find something "division like" with the properties they're interested in to be relevant to the question (even if it is missing some other properties that most mathematicians consider to be essential to the notion of division).