IANAL - but it seems to me that in the link's scenario, you're not amending that contract. You're ignoring or rejecting the initial proposed contract (including this amendment clause) and instead proposing another (different) contract. Since you never agreed to that original amendment clause in the original contract, how would it have any force in the first place? Naturally, if you were to attempt to modify the contract _after_ both parties signed, you would need such a written document executed by all parties.
> This Contract can only be amended through a written document formally executed by all parties.
To me, that suggests the contract can only be amended if all parties provided a written document approving the amendments.
In this case, only one party provided the required documentation.