@cam welcome to the Atlassian developer community.
REST APIs between Jira Server and Jira Cloud are similar. They do have a shared history (until about 5 years ago). However, Atlassian makes no guarantees these are compatible. V2 Cloud is not the same as V2 Server (but yes, there is a V2 in both). The list of differences is too long and too difficult to maintain. However, the major differences would be identity, editor, and configuration. V3 is even more different, with different endpoints, and payloads. I don’t know that you could do something like /rest/api/latest, without pulling a lot of complexity into the code that handles those payloads. I think it is easier to handle as if they were different APIs.
The key to understanding JSM in both Server and Cloud is platform vs product. For example, issues (like your example) are Jira platform (link to Server docs). But queues, SLAs, and customer requests are JSM (Server docs again).
All that said, there are Marketplace partners who do manage both Server and Cloud with same code base. For recent example, consider:
Dev and testing can be done 2 ways. You can be (or become) a customer. Dev licenses are to customers. Or, you can use the Atlassian SDK to run JSM, with limits (like 3 days between reboot).