When you own bitcoin, what you own is a key that gives you access to a bitcoin credit on the bitcoin blockchain thats assigned to your private key.
Nope. When you "own bitcoin", what you own is a private key that allows you to provide a digital signature that meets the requirements of specific transaction outputs in order to be allowed to use them as transaction inputs.
The key allows you to transfer some or all of your credit directly to another key.
There is no "credit", there are just outputs with requirements. Meet the requirements of the outputs and you can list them as inputs. Then you can create new outputs with their own requirements.
When it's turned into fiat, what does the blockchain record?
It isn't "turned into fiat". It is exchanged for fiat. When gold is "turned into fiat" what does the gold record? When you purchase a car, and the car is "turned into fiat" for the seller, what does the car record?
Not that ownership is transferred to a different private key. Does it record an ID of the exchange that converts the currency?
A new output is created that requires a signature from the bitcoin recipient if they want to use it as an input. In exchange for creating this output, the recipient gives you fiat (or whatever else you both agree to exchange).