Google Natural Language key?

To use the Google Natural Language Connector, we downloaded the private key as JSON.

It says this "Copy the key and enter in the Settings menu inside the connector".

there are many fields in that JSON, what is the exact one and format to get it to work?

It appears the steps have changed. You need to create an API key, see updated instructions on the Google Natural Language documentation page:
https://seotoolsforexcel.com/google-nl/