Thanks for your detailed response Lauren.
FWIW, I think buying your house is better in most, but certainly not all, typical circumstances.
As for what you quoted from my article, first, I think there's a typo in the first bit, that should be "owe" rather than "own."
As for the second part, I fully stand by it. As I wrote in my article, your house is an asset. Period. If you have a mortgage on it, that mortgage is a linked liability.
If you do have such a mortgage, the bank does *not* own your house. They "own" the mortgage. As long as you keep making timely payments, they can sell the mortgage but they can't sell your house.
The critical part, of course, is that non-trivial "as long as you keep making timely payments." You may think of it as semantics, but when you're talking about contracts, words matter, a lot. To get the bank's money to help buy the asset (your house), you agree to give them certain of what would normally be only your rights in the property.
Specifically, that you can't sell the asset without paying off the loan, you must keep it insured, you must keep the lender on your homeowner's insurance as the lien holder, you can't displace the bank as the first lien holder, etc.
I hope that clarifies things somewhat.
Regarding your decision tree's simplicity, I did acknowledge that it was necessary, and just noted that this simplicity makes it less useful for any specific circumstance.