Devices
Your plan works on one machine at a time. Connecting Butter Motion on a new computer signs the previous one out, and there's nothing to disconnect manually.
The one-device rule
When you connect a new device, the one that was connected before is signed out automatically. You'll pick up right where you left off on the new machine. The old one simply stops being able to generate until you sign in there again.
The screen you use to connect a device says as much: "One device at a time, the previous one signs out."
On the computer that got signed out, the panel tells you why: it shows "Signed in on another computer" with a Use this computer button. One click opens the browser, you confirm this machine, and it becomes the connected one again (the other computer signs out in turn).
See and manage connected devices
Your account page has a Connected devices section that lists what's currently signed in. From there you can use Sign out all devices to clear every device at once, handy if you've lost a machine or just want a clean slate.
Signing out of the account area in your browser (Sign out on this device) is not the same as disconnecting a plugin device. That's done here, in Connected devices.
How long a device stays connected
A signed-in device stays connected for 30 days. After that it signs out on its own and you connect it again through the browser, the same quick sign-in flow as the first time.
Connecting a device and signing in are the same step. Signing in, normally inside the Butter Motion app during install, is what connects the device. If you need to reconnect later, the panel's own Sign in via browser button does the same thing.