If I go back to your first email, it appears you paid before they told you that you were on the waiting list, so I'm not sure what this woman's argument is.
Either way, in terms of your case, I think this is simple contract law. Contract law basic principles are that there is an offer and an acceptance, and those two together make a contract. You have to have both. Here we have two potential offers and two potential acceptances:
Offer 1: Via letter and email - I offer you a spot in this school upon payment
Acceptance 1: You pay and accept their offer.
If this is how a court sees the facts, then the school breached your contract because they did not give you a spot, and owe you damages (which in this case you are saying is simply a refund). It does not matter that they later offered you a spot, because once you knew of their breach you took appropriate measures to get into a different school. Your damages would either be for them to pay the other school's deposit and let you enroll in their school, or refund your deposit at their school.
Their scenario: The school admits, yes, we made Offer 1, but we rescinded it (took it back before you could accept it), and gave you:
Offer 2: Put you on the waitlist in exchange for payment.
Acceptance 2: You pay and accept their offer.
The above is what they are trying to argue. This is of course an oversimplification of how contract law works, but from a reading of your facts, the cannot argue they properly rescinded their offer and gave you a new offer, because their original offer had a deadline that you met, and also, you paid before they gave you the new offer of a waitlist spot.
Since this is a small claims case you probably don't need a lawyer, I just don't see from these facts how they can take your money.
By the way, since you are at a new school anyway, what is the harm in telling us the name of this school? It's quite possible parents of kids who attend that school or even school faculty read Geobaby and would see this post and help remedy it.