The article doesn't propose anything especially different from Lamport clocks. What this article suggests is a way to deal with non-idempotent message handlers.
Its true that idempotency can sometimes be achieved without explicitly having message identity, but in cases it cannot, a key is usually provided to solve this problem. This key indeed encodes the identity of the message, but is usually called an "idempotency key" to signal its use.
The system then becomes idempotent not by having repeated executions result in identical state on some deeper level, but by detecting and avoiding repeated executions on the surface of the system.
> Idempotency is a property that says the handler can run as many times as I like
.... using some input. If that input has a key and the handler bails on execution, your definition is not at all violated. Its only violated if you don't consider the check a part of the handler, which is an arbitrary limitation.
Regardless, I think your interpretation that avoids identifying the message explicitly leaves a very narrow set of idempotency candidates, which is not a useful definition. Under that definition, really only reads are idempotent, as any state setting can be later retried and give a different result if other state settings are interleaved.