Your definition of what you
think a private key is, is irrelevant.
If you import your bitcoin private key on an hypothetical platform that gives you access to all functionality of that private key via a self generated api key to your account on the platform, would in that case the api key act like a private key ?
YES or NO ?
An API key isn't a private key, because I cannot import it in Bitcoin Core and spend my
BTC.
It could match your description only indirectly, by extension; because the API key that was given to you allows you to manipulate the cryptographic private key that you gave in exchange of the API key.
Also, you are shortcutting the fact that this API key was given to you by a privately owned third party entity and this entity does the manipulations around the cryptographic private key on your behalf.
This isn't really among the "best practices".
To be clear, strictly speaking a cryptographic private key comes in a pair, with a public key.
So, unless the "API key" is mathematically linked to another set of data that has been generated by a cryptographic signature algorithm, no, this cannot be called "private key".