If you boot to a menu, showing Disk Utility, and Reinstall macOS, and a couple of other choices, and not to an actual desktop, with a dock, etc, then you are just booting to a recovery system, or even a bootable installer. You can use that to reinstall macOS as a full system, but it is not a full system, per se. Another item that you will have is a Utilities menu, with probably 3 items: Terminal, Startup Security utility, and Network Utility.
How large is that external drive? is it a USB flash drive (thumb drive), or an actual external storage with (at a minimum to be usable) 120GB of capacity?
If you are booting to an installer (which might be only 8GB size volume), then you just need to have a drive with enough space for a full system install. That's where you can use that drive, if it is 120GB or more.
You can use that Disk Utility to see what drives are there, and how large those drives might be.