No, E2EE doesn't mean it's encrypted until the service provider decrypts it. E2EE means the service provider is unable to decrypt it. What you are describing is encryption in transit (and possibly at rest).
Bank data is never E2EE because the bank needs to see it. If banks call it E2EE they are misusing the term. E2EE for financial transactions would look like e.g. ZCash.
I would argue it depends on context. E2EE means it's encrypted until the "target" receives it. For a messaging protocol, it's the intended recipient of the message. For what the person you're replying is discussing, the intended recipient IS the bank.
That being said, the person you're replying to seems to be saying that "the server" is always an "intended" end, which is wrong.