In the first, a global adversary (say, NSA) records all your interactions via a chat service. This chat service does not use PFS. At a certain point, they capture the private key from one of your devices that uses the service, and is able to decrypt all your messages.
In the second, the same global adversary records all your messages, but this time from an otherwise identical service that DOES use PFS. The adversary captures your private key but can only use it to MitM attack you going forward. However, before discarding your device, they check your conversation history -- because your diligent backup and transfer of all your conversations since you signed up for the chat service, they now have all your messages. PFS did not help.
Now, you could change scenario #2 to where you don't have an option to back up and transfer message history (or simply choose not to), and that's essentially Signal. In that case, the adversary is pretty much hosed except for the messages your device just happened to have on it. Adversary steals message backup == adversary steals private key w/o PFS
I'm no crypto expert but that "later date" when talking about PFS is to avoid an external dragnet recording all your ciphered streams and then deciphering them once they have your non-PFS secret key.
I mean, in your definition basically all the messages should be ephemeral on your device and on each recipient device to have PFS.