Your net worth is your total assets less liablities. If you have $1m in cash today you are a millionaire. However, the statement that if you buy a house tomorrow for $1m in cash, you are no longer a millionaire, seems a little ambiguous.
The flip side is, if you include a property, should you include your cars, antiques, pictures, jewelry? If you exclude your main residence, do you exclude your holiday home that you rent out and that you bought as an investment/second home. Some people buy pictures, antiques and jewelry as items that they make use of, but that can be liquidated in times of need, and hopefully at a profit.
It really boils down to who wants to know, your bank, your buddy, or the local friendly burglar.