elusian How do I change the terminal there?
It's been years since I've used Budgie, and I couldn't remember a way to change the default application for the terminal, so I fired-up a Budgie VM and searched for a way. I still couldn't find anything that might help. Budgie is a pretty DE, but not very configurable, which is the main reason I stopped using it. If I've missed something, perhaps someone else will help.
Other DEs make it easy to substitute a different application for one that's commonly used. Xfce, for example, allows me to substitute Terminology - the most configurable terminal emulator I've ever used - for the default Xfce terminal. In the settings, it's right in the Default Applications section, where you can change any of four common defaults.

And in Plasma settings, it's found in the System Settings, also in the Default Applications section. True to Plasma's nature, it's far more configurable. This is the DE I actually use, and have for years, and its configurability is the main reason why I do.

I wouldn't recomment the GNOME DE to anyone, but it seems to be the same as Budgie. You can add another terminal emulator by simply installing it, and it then shows up as an application in the UI, but I couldn't find a way to change the default.