Neither of these mention networks, only protocols/schemes, which are concepts. Cryptography exists outside networks, and outside computer science (even if that is where it finds the most use).

This is ridiculous rules lawyering and isn't even done well. Such schemes inherently assume multiple communicating parties. Sure you might not need to have a network but you still have to have distinct devices and a communication link of some sort (because if you have a direct trusted channel you don't need cryptography)

You're also wrong about your interpretation.

Here's how to read it:

At point A both parties create their long term identity keys.

At point B they initiate a connection, and create session encryption keys with a key exchange algorithm (first half of PFS)

At point C they exchange information over the encrypted channel.

At point D the session keys are automatically deleted (second half of PFS)

At point E the long term key of one party is leaked. The contents from B and C can not be recovered because the session key is independent of the long term key and now deleted. This is forward secrecy. The adversary can't compromise it after the fact without breaking the whole algorithm, they have to attack the clients as the session is ongoing.

This is motivated for example by how SSL3.0 usually was used with a single fixed RSA keypair per server, letting user clients generate and submit session encryption keys - allowing a total break of all communications with the server of that key is comprised. Long term DH secrets were also often later used when they should be single use. Then we moved on to ECDH where generating new session secrets is fast and everybody adopted real PFS.

Yes compromising the key means you often get stuff like the database too, etc. Not the point! If you keep deleting sensitive data locally when you should then PFS guarantees it's actually gone, NSA can't store the traffic in their big data warehouse and hope to steal the key later to decrypt what you thought you deleted. It's actually gone.

And both of the above definitions you quoted means the same as the above.

In any case, both of these scenarios create an attack vector through which an adversary can get all of your old messages, which, whether you believe violates PFS by your chosen definition or not, does defeat its purpose (perhaps you prefer this phrasing to "break" or "breach").

Playing loose with definitions is how half of all broken cryptographic schemes ended up insecure and broken. Being precise with attack definitions allows for better analysis and better defenses.

Like how better analysis of common attacks on long running chats with PFS lead to "self healing" properties being developed to counter point-in-time leaks of session keys by repeatedly performing key exchanges, better protecting long term keys by for example making sure software like Signal make use of the OS provided hardware backed keystore for it, etc. All of this is modeled carefully and described with precise terms.

Edit: given modern sandbox techniques in phones, most malware and exploits doesn't survive a reboot. If malware can compromise your phone at a specific time but can't break the TPM then once you reboot and your app rekeys then the adversary no longer have access, and this can be demonstrated with mathematical proofs. That's self healing PFS.

Anyone can start a forum.

Fair point, but my cryptography forum (reddit.com/r/crypto) has regulars that include people writing the TLS specifications and other well known experts. They're hanging around because the forum is high quality, and I'm able to keep quality high because I can tell who's talking bullshit and who knows their stuff.