Correct, C2C it’s the abbreviation of Cloud-To-Cloud. You can see several references to this abbreviation in the ST Schema documentation.
About the test of Health Check. You only need to make the test for the corresponding protocol because it works differently in each one of them. In the case of C2C, LAN, and MQTT, the developer has more control over the connection. For example, if the device goes offline unexpectedly, it might be an issue with the Access Token on your side because the Schema makes a periodic discovery and thus stateRefresh, if it gets an error using the token to authorize, you’ll get a global error in your Schema Connector and the device will me marked as offline.