The difference is that for solid-state logic, the heat is purely a waste byproduct, whereas for thermionic valves, the heat is the energy used by the electrons to move between the electrodes, hence the name. If the valves are cold, you have to wait for them to warm up before they can do anything.
All the same, "cold boot" doesn't refer to letting tubes cool down. If power is restored while the tubes are still hot, it's still a "cold boot".
The electrons are not moved by heat; they are moved by the electric potential between the electrodes. The heat goes into maintaining the cathode at a certain temperature, at which the electrons "boil off" in sufficient numbers, but that requires only a smidgeon of the energy. Though depending on it, tubes are incredibly wasteful of heat.
So you say silicon and other parts don't require heat? Check that data sheet again! If the part's operating range is from -40°C to 70°C, then that part requires heat. If the ambient temperature happens to be -100°C, you have to heat it, perhaps by installing a heating filament into the chassis.
E.g. check the TI datasheet for the NE5532 op-amps:
Free air operating temperature:
NE5532, NE5532A: 0 to 70°C.
SA5532, SA5532A: –40 to 85°C.
(That standard part is quite delicate there, only down to zero!)
Conversely, thermionic tubes for a hot environment could be built without heating filaments. If the ambient temperature is 500°C, and that happens to be quite enough for that tube's cathode, then it Just Works, like your NE5532 IC at 3°C.
Is this known to work? I would have thought that running a whole tube at high temperature would cause the grid to emit thermal electrons, creating a current between grid and plate. Even though the grid doesn't have the low-work-function coating that the cathode does, its surface area can be larger and it's usually more negative than the cathode so I'd expect substantial currents.
If this does work, you could save a lot of power by building heated, insulated enclosures for entire tube appliances rather than heating each cathode and cooling every tube.
"Conversely, thermionic tubes for a hot environment could be built without heating filaments. If the ambient temperature is 500°..."
There's a lot of physics and engineering behind thermionic emission in a typical vacuum tube and the subject shouldn't be overlooked just because tubes are less commonplace than they once were.
Until the transistor became popular in the latter half of the 20th C. essentially all of electronics relied upon thermionic emission so much effort went into understanding it. Moreover, it's still very important in specialized areas such as in scientific instruments, high power transmitting tubes, TWTs (Traveling Wave Tubes), CRTs, etc.
Thermionic emission obeys Richardson's Law—after physicist Owen Richardson who received the Nobel Prize for his work in 1928, and its study is an essential part of thermodynamics: https://en.m.wikipedia.org/wiki/Thermionic_emission.
When I was learning about this stuff I had a little book titled Thermionic vacuum tubes and their applications by Sir Edward Appleton (who is more famous for his wartime research into radar). Still, for anyone who's interested in thermionic emission then this could be a useful reference.
I'd add that when it comes to practical methods of manufacturing thermionic cathodes, info on low work function oxide emitters etc. then there are references that are better or more practically orientated (but offhand I cannot remember the names of any).
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}.)
