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.
No, it doesn't depend on context. The intended recipient of a financial transaction is not the bank. The intended recipient is the party you're trying to pay. It is possible for financial transactions to be E2EE and completely indecipherable by anyone but the two parties of the transaction. Crypto like ZCash can do it. Banks cannot.
Can you expand on this a bit. It was my understanding that you're telling the bank to pay the vendor (from your money/credit). In that case, the bank certainly needs to know about the transaction... so it can make the payment.
Are we talking about 2 different things here?
I suggest researching how ZCash uses zero-knowledge proofs to allow paying money from your balance to another person's balance without any middleman like a bank being able to decrypt your transaction, while still allowing everyone to verify that important invariants are maintained, such as not allowing you to spend more money than you have.
This is what it takes to make a financial transaction E2EE. I'm not saying that banks could or should do this. I'm just saying that their systems do not qualify as E2EE unless they do. It's not ambiguous.
While what you're saying makes sense, it's not the normal use of the term - in fact, the term 'end to end encryption' was basically coined to differentiate user-to-user encryption (through an intermediary service that can't decrypt the message) from the regular case (user to service encryption) that you're talking about!
It wasn't coined, it was reused. It historically meant things that were encrypted from the client to the server, e.g. SSH, SSL, TLS, etc.
RFC 4949 (Internet Security Glossary, Version 2) from 2007: https://datatracker.ietf.org/doc/html/rfc4949
$ end-to-end encryption
(I) Continuous protection of data that flows between two points in
a network, effected by encrypting data when it leaves its source,
keeping it encrypted while it passes through any intermediate
computers (such as routers), and decrypting it only when it
arrives at the intended final destination. (See: wiretapping.
Compare: link encryption.)
Examples: A few are BLACKER, CANEWARE, IPLI, IPsec, PLI, SDNS,
SILS, SSH, SSL, TLS.
Tutorial: When two points are separated by multiple communication
links that are connected by one or more intermediate relays, end-
to-end encryption enables the source and destination systems to
protect their communications without depending on the intermediate
systems to provide the protection.