I understand that you have some questions about the Slingbox and how it works.
According to your current scenario, your best option is the Slingbox PRO-HD. It has an internal tuner that allows you to "split" the TV signal without affecting to the home viewers. The only disadvantage is that when the home viewers watch the signal that comes from the set top box, you will only be able to watch the coaxial signal. This means that you will only get the basic channels signal.
On the other hand, there are also another option: if you get either the Slingbox PRO-HD or the Slingbox SOLO, you can use a "dedicated" set top box to hook it up to your Slingbox. This way, you can get the connection to your Slingbox without affecting any one else. In all case, I recommend you to check the following links in order to get all the information about the Slingbox:
Hope this helps!