Does Claude Code Desktop actually use the terminal/SSH to deploy changes, or is it a separate interface?

Yes. Claude Code Desktop is confirmed to use the terminal under the hood - it's described as a hybrid of a prompt interface and a terminal running simultaneously. To deploy, you give it access via an SSH key (e.g., for a Digital Ocean server), and it can execute deployment steps itself. One member noted that after doing the deployment steps manually a few times, they had Claude build a 'skill' to automate the repeated three or four deployment steps. If Claude needs something it can't do directly, it will prompt you to log into a website, and often once told 'do it for me' it can complete that step too. It's viewed as functionally similar to using terminal directly, just more user-friendly.