And then there was Reagan's "My fellow Americans, I'm pleased to tell you today that I've signed legislation that will outlaw Russia forever. We begin bombing in five minutes."[0] I was in Moscow when that went public. The Soviet government was not amused.
Removed from that quote is context of course, which is extremely relevant, but that didn't stop the media from making sure they induced a profiteering panic among the people that subscribed to them despite the outcome being possible nuclear death.
Well, of course, he was joking. Just a sound check.
I have no clue who leaked it, in what order. From a reprint of contemporaneous coverage, I gather that many technicians heard the remark.[0] And eventual news coverage mainly served to put rumors at rest.
There's some interesting videos on YouTube from people who used satellite dishes to grab unencrypted transmissions that weren't intended for the public (although I don't know if that was the case here). Things like reporters talking shit about guests during a commercial break (since commercials were spliced in later on), Bill Clinton getting his makeup done, etc. There's an interesting documentary/compilation at [1] although in my opinion it's a little overly dramatic.
Edit: Also, Charles Stross' A Colder War.[1]
0) https://en.wikipedia.org/wiki/We_begin_bombing_in_five_minut...
1) http://www.infinityplus.co.uk/stories/colderwar.htm