Yeah, joseph529, it's still a computer that strangers use. The "remember this device" option saves a cookie or a token on that specific machine. If they don't wipe it right, or if there's malware, your login could be sitting there for the next person. Your brother gave you solid advice.
Wait, you're telling me you do personal stuff on a library computer? Don't they wipe those things clean after every single user? How is your info even still on there? That seems like a massive security risk they should have locked down.