Rather than have to remember which operating system is running at the time, I decided to forcibly make Windows use a different MAC address. This means that I can now assign each machine a different hostname in DNS. Checking which operating system is running is now as simple as pinging the hostname corresponding to the operating system.
You can change this is both Linux and Windows, but I decided to keep my main operating system using the machine’s original MAC address. Changing this in Windows involves editing a value in Device Manager: