Does the Current Sharepoint Online REST API Work With ANY Other Versions of Sharepoint?

The following (current/2018) REST API works great with the Microsoft hosted version of Sharepoint (https://yourcompany.sharepoint.com):

https://learn.microsoft.com/en-us/sharepoint/dev/sp-add-ins/get-to-know-the-sharepoint-rest-service

Does anyone know if this SPECIFIC API also works with/supports SharePoint 2016, 2013, and/or 2010?

Solution

That set of operations should work with Sharepoint 2016. SharePoint 2013 and SharePoint 2010 have its own set of REST operations. Of course the methods have evolved but the apis are quite similar. For example SharePoint Online (and on-premises SharePoint 2016 and later) REST service supports combining multiple requests into a single call to the service by using the OData $batch query option, but on SharePoint 2013 on premises batch was not supported.

In this link you'll find the list of operations and samples for the SharePoint 2013 REST API:
SharePoint 2013 - REST API reference and samples.

Please watch the video in the link as it's a good introduction on how the API evolved. One thing to notice is for example how the old WCF "REST" api changed from
- http://contososerver/_vti_bin/client.svc/web
to this one
- http://contososerver/_api/web

Here are some links for the documentation about the SharePoint 2010 "REST" API:

SharePoint 2010 - Using the REST Interface: How REST is implemented and some examples
List Items manipulation via REST API in SharePoint 2010: Basic example
SharePoint Foundation REST Interface: List of APIs that support REST

So to finish answering your question, that specific link wont work for SharePoint 2013 and SharePoint 2010 (Some examples might work, but depends on the operation/method)