All of our applications will connect to a Slingbox when you're not on the same network as the Slingbox. It sounds like you want to watch your Slingbox on a TV that is not located with the Slingbox itself. Here is an article that covers the ways you can do this at length:
What was probably meant by "needing to be on the same network" is that in order to view your Slingbox using AppleTV, Roku, or Chromecast, you need to have a mobile device with our application installed to be on the same wi-fi network as it.
Hope this helps!
EDIT: By the way, I noticed on your account that you have a Slingbox SOLO. Of the methods to watch on a TV, only the AppleTV and the Roku are supported for use with this model Slingbox, so if you're looking to stream the SOLO out to your vacation home, you'll want to look into one of those two. You can find information about this here:
I read the article you linked, and you really didn't answer my question at all. I tried to be as specific as possible. Clearly Roku and Chromecast won't work for me unless I'm reading it wrong, it seems like the Fire TV will work for me. I don't have a smartphone or a tablet and I'm in a different location.
Is the Fire TV my only option to watch my slingbox on a TV?