Around the dinner table the conversation is lively
I think there may be a clue in the grammatical structure of the sentence. The text is also accurate written as "The conversation was lively around the dinner table". its one of those perfect sentence structures that can be split and reversed while still carrying the same meaning. I feel like there is some significance to how it is written, not so much what it says.
I think the message must refer to a moment many years ago when Cooper recorded that sentence into his dictaphone. He could have been having dinner at an FBI event or something and took a break from the table to record some thoughts into his dictaphone. It might not actually mean anything significant in itself as a message, but might just be a way for Mr C to let Diane know that it's him and he's out there.
Ian Cook: Some people suggested on page 1 of this thread that the difference in the text that Dopplecoop sent, versus what Diane received, may be because Diane wasn't Dopplecoop's intended recipient--whoever Dopplecoop sent the message to, may have re-sent it to Diane, maybe without Dopplecoop's knowledge or approval, and whoever that person is, they re-entered it differently into their own phone.
Here's the cache of the page that appeared July 10 2017 at www.ihatecommercials.com (the original page is no longer online), in which "doktorteeth" posted the two versions:
around the dinner table the conversation is lively
AROUND THE DINNER TABLE, THE CONVERSATION IS LIVELY.