In our Jira Server platform REST API reference
We have HTTP code and responses.
Which applies to the responses you get in the UI.

I.E: For non-existent issues:

You can't view this issue
It may have been deleted or you don't have permission to view it.

The same error message is not shown when using REST API endpoint:

https://<Jira_Base_Url>/rest/api/2/issue/{Issue-Key}

curl -u admin:admin http://localhost:49124/j9124/rest/api/2/issue/TEG-12345
{"errorMessages":["Issue Does Not Exist"],"errors":{}}%  

We should update our documentation. to show all available responses we have in Jira.