Material implication can be thought of as a kind of very simple, special case of a conditional. It is a truth function, which is to say that its truth value depends only on the truth values of its antecedent and consequent, not on any other semantic connection between them. It serves only to express a sufficient condition between the truth of its antecedent and the truth of its consequent. It does not cope with common features of ordinary English conditionals, such as: they might be uncertain, they might hold by default under normal conditions but have exceptions, they might be implicitly quantified, they might serve to imply stronger relationships between the antecedent and conequent such as a causal or evidential relation. Also, we can conditionalise all kinds of speech acts, e.g. questions, commands, offers, threats, bets, promises, etc., which obviously do not have truth values.
In your examples, I would say that the mathematical example does work as a material implication. Indeed, mathematics is a domain where material implications work well because mathematics is not typically concerned with uncertain claims or unstated exceptions. "If John is in school then his bag isn't in the house" could be a material implication at a pinch, but it fails to capture the fact that this might have exceptions: maybe John forgets occasionally, or is taking part in a school event and doesn't need his bag. Also, it skirts over the fact that there is a connection between things: we are given to understand that John's bag isn't in the house because John takes it with him to school. I would say your last example is a conditional promise, not an implication.