Why don't we try a different approach?
I think this and that are separate issues.
I own two houses. I bought this place about three months ago and I own another place down in the valley.
The の is there to indicate that it is the difference of something. For example this is like saying "the separate ______" or "the different _____" or "another ______". 別 by itself just means separate, different, or another.
The reading is the same as the one you learned. Move on to a 別の漢字, mmk?