I fear we are going around in circles now.

But one more time. You can't sign a valid transaction spending my output using my public key. It would require my private key. You can sign a valid transaction spending your transaction with your private key. All you can do with my public key is add it to your signature spending your output, obscuring the fact of which output was spent. This is just obscuring your spend with my public key, not spending my output.

In both cases, a foreign public key can be added but you can't construct a valid transaction without the private key.