Thanks Chris.
For documentation and so that others might be able to solve the issue a bit quicker. With an additional header, the API call through curl is now working:
curl -X POST 'https://tables-api.softr.io/api/v1/databases/<MY-DATABASE-ID>/tables/<MY-TABLE-ID>/records' \
-H 'Softr-Api-Key: <MY-API-KEY>' \
-H 'Content-Type: application/json' \
-d '{ "fields": { "<MY-FIELD-ID>": "test" } }'
Maybe someone wants to add this to the official database API documentation? The code examples provided don’t feature this header.