The company made it clear in their most recent communication that they were expecting you to stay one night and you stayed two. In this case, I think you are pretty clearly in the wrong.
Normal business travel etiquette is to fly out the night before and leave the last day business is conducted. If you're being reimbursed, it's on you to communicate any deviation: You want to tack on a personal trip to the beginning or end, the meeting ends such that same-day travel isn't an option, you are staying with friends and don't need a hotel or car, etc. Any added costs are your responsibility (The flight out on Friday is $100 and the flight out on Sunday is $500, you pay $400.). Depending on company policy, any savings may be yours as well (A hotel is typically $150 but you're staying with family, you may be able to claim that $150 or expense $150 in host/hostess gift, meals, etc.).
Revise your reimbursement request to subtract a night at the hotel, any additional car rental costs (including gas, if reimbursed), any food or drink for the 3rd day, etc. I don't know of any way to look up historical flight prices, but hopefully a flight out the following day wasn't substantially more expensive. However, if the company is able to provide evidence that the flight was more expensive, reduce your reimbursement request accordingly.
Provide the new request to the company and apologize for your mistake.
There are two mistakes that led to your current predicament.
You paid for your own travel. Why in the world is the company not scheduling and paying for the airfare and lodging up-front? Expecting a potential hire to float hundreds of dollars just for the privilege of interviewing is a good way to scare off candidates. Any company/opportunity that requires you to put down money up-front has a high likelihood of being a scam, though it doesn't sound like your current situation is a scam. If you get any money back, consider yourself incredibly lucky and never make this mistake again.
You and the company were not transparent about your expectations for what this trip entailed. The company likely expected you to fly out in the evening after the interview, but they didn't communicate this expectation to you (or they did and it's your mistake). This causes an issue of an extra hotel night and possibly different airfare costs for the next day. You didn't properly communicate with the company to learn their expectations and/or understand the reimbursement policy.