Rent is money out of your pocket. So is renting a "liability" under your strange definitions?
Its actually an expense, whereas the cost to carry a mortgage is interest expense. Your mortgage is a liability, and your home is an asset. These are basic accounting matters, but cling to your definitions if it makes you happy.
and in accounting the house is a so-called asset, but it's money out of your pocket and thus it "strangely" actually a liabiliy cuz mortgage and taxes are attached to that property