The meanings we give them today go back almost 50 years.
Then the trail runs cold, pardon the pun.
According to Google n-gram searches, "cold reboot" and "warm reboot" didn't exist at all until the late 1970s.
Related terms like "cold boot" and "warm boot" come up, but only in references to footwear. Can't find any computer uses 1950-1970.
The "cold start" and "warm start" mostly come up in automotive, aviation or marine contexts, confounding the search. Likewise, not finding computer uses.
Not finding computer uses for "cold restart" and "warm restart", either.
All these terms take off as computer terms in the late 1970's.
By the way, an important feature of the warm reboot is (possibly) that data is still in memory (if you have the kind of machine whose memory is wiped when power is cut, not magnetic core). In machines with simple operating systems, you could recover your program or data from memory after a reset.
(I think the timeline in the n-grams coincides with the explosion in microcomputing and reflects that: penetration of the jargon into mainstream writing. Nevertheless, the case doesn't seem to be good though for a vacuum tube origin of warm {start/restart/reboot}.)
