Now, I just want to know how they work in a technical level? For implementation of something like those websites, I can simply use FixedFloat API and limit it to some certain currencies, but I just want to know how they work in their core. Do they buy my desired coin from a CEX in their wait time? Do they rely on people who want to do quick exchange? or do they send requests to a lower level API or service for doing the swaps?
Most probably you are talking about instant swap, they aren’t non custodial exchanges. The liquidity holding by the swap or exchange, if you looking for a noncustodial exchange then it’s called decentralised exchanges. So the instant swap collect their own liquidity from various exchanges or other sources. When users place swap orders they complete it from their liquidity.
There are many exchanges that use liquidity from some other exchanges or platform as well through API. So you have to read their terms to know how their swap have been working. Also you have to understand different between centralised exchanges and decentralised exchanges as well. Remember, most of the instant swap or exchange are centralised, they are controlled by a company.