This happened to me and I found out that the prompt changes to the name assigned by your DHCP server so will vary depending on what network you are connected to. So like you say its a router thing so if you want you could change it there rather than change it within OS X.
Another method rather than using the terminal command is to turn the feature off in Sharing Preferences, where you enter your computer name there is an edit button with a tick box 'use dynamic global hostname'
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