Hi,
Indeed to deploy an API service, you will need to make a dedicated "API node" installation, as documented in https://doc.dataiku.com/dss/latest/apinode/installing-apinode.html.
Then you can use your "Design node" as an "API Deployer" to push API services to production on the installed API node. In a large team, you would probably configure a standalone API Deployer node that multiple Design nodes can push to. See this documentation for more details: https://doc.dataiku.com/dss/latest/apinode/installing-apideployer.html#using-a-standalone-api-deployer.
Note that the "Design node" contains an internal "development API server" for simple testing of API services. But it cannot be used for real production testing and deployment.
Hope it helps,
Alex