Looks like a JsonDocument doesn't work either

Yeah, that’s how they defined it in specs: https://github.com/Azure/azure-rest-api-specs/blob/master/specification/logic/resource-manager/Microsoft.Logic/stable/2019-05-01/logic.json#L6439

An object with arbitrary properties (so, a dictionary in C#)

OK I've managed to get this to deploy with all of the correct swagger etc. I am struggling to get the connection parameters section of the deployment to work though (i.e. the equivalent to line 18 here) This is the relevant code from the original ARM template

"connectionParameters": {
  "api_key": {
    "type": "securestring",
    "uiDefinition": {
      "displayName": "API Key",
      "description": "The API Key for this api",
      "tooltip": "Provide your API Key",
      "constraints": {
        "tabIndex": 2,
        "clearText": false,
        "required": "true"
      }
    }
  }
},

I can't see how to set the UiDefinition section when the type is Securestring (as opposed to OauthSetting).

As far as I can see there isn't a way to achieve this and it prevents me from being able to edit the API Key for any connections based off of the new connector. A workaround is to deploy the connector then go in and edit it, specifying that it needs to use an API Key. This populates the missing uiDefinition and allows everything to work. Somewhat defeats the purpose of automated deployment though :(