What you explained is the workings of a custodial service. The deposit addresses your users deposit to are under your control. You own the keys to them. It's exactly the same as with other instant exchanges. For instance, eXch, which was very popular on Bitcointalk. After reading your T&Cs and the mention of sending customers private keys, I thought that was the case with all swaps, but as you explained below, it isn't. Exchange owners seem reluctant to call their service custodial. You shouldn't be if your intentions are good. Your service isn't trustless and it's custodial but you can build a brand here if you do it right. There is a market to fill. I wouldn't give custody any importance in your OP. Most of us here know how instant exchanges work and someone could see it as annoying when claims are made that aren't correct. 

You're correct in that you'll have to trust us for a few minutes when we process and perform the exchange. We acknowledge this and we are ready to earn users' trust with time.

the terminal design you are going for is nice and all, but the swap field is trash in all honesty, took me a bit to find it. you should change it to make it more obvious.

i agree with @Pmalek here.

calling your service "Non-custodial" is dishonest and misleading. to put it simply... if i'm sending my coins to an address i don't control for a swap, it's custodial no matter how you try to spin it. i suggest you drop the "Non-custodial" nonsense.
your service is aimed at people who care about their privacy and know better than that, so making false claims like this will do more harm than good for your brand.

edit: i tried to use the website with javascript disabled, but that seems to break it, if you could make the website javascript free, that would be great.

What you explained is the workings of a custodial service. The deposit addresses your users deposit to are under your control. You own the keys to them. It's exactly the same as with other instant exchanges. For instance, eXch, which was very popular on Bitcointalk. After reading your T&Cs and the mention of sending customers private keys, I thought that was the case with all swaps, but as you explained below, it isn't. Exchange owners seem reluctant to call their service custodial. You shouldn't be if your intentions are good. Your service isn't trustless and it's custodial but you can build a brand here if you do it right. There is a market to fill. I wouldn't give custody any importance in your OP. Most of us here know how instant exchanges work and someone could see it as annoying when claims are made that aren't correct. 

Got it, thanks. But why only for DAI and not also for ETH or all cryptos for that matter?

Yeah, and that is what makes it centralized and custodial, just like other instant exchanges, the moment funds are deposited into the deposit address you provide, you have custody of the coins. I don't have a problem with instant exchanges, in fact, i believe we need more of them that are legitimate and privacy oriented, but it is good to set the records straight that they are centralized and custodial, so newbies know this before going ahead to use them. Welcome and Good Luck.

1. "Non-custodial" - This term is used as a contrast to the process of regular CEX exchanges where users deposit their funds into a custodial wallet owned by the exchange, perform the swap in the exchange, and have the output currency in a wallet owned by the exchange, before they're able to withdraw into their own wallet (if they even do that).

2. "If you refuse my exchange, the website crashes, or something else happens, how do I get my coins out without your help?" - You're correct in that you'll have to trust us for a few minutes when we process and perform the exchange. We acknowledge this and we are ready to earn users' trust with time.

3. "Transmitting private keys to customers" - Currently this is only when we refund DAI -> X exchanges. If for some reason we are unable to perform the requested DAI -> X exchange, we would just transmit the private key of the receiving DAI address to the user (To simplify the process, so we would not have to 1. ask users for their refund address 2. send eth to the receiving address and 3. send the DAI to the refund address).