coldboot is not an exploit name, it's a state of the boot process.
it means booting the console from "cold" state, shutdown state > power > boot !
in opposition to softboot (reset the device without shutdown)
if an exploit is "coldboot" it means that it's applied automatically right at console boot, and you don't have anything to do to get CFW.
Haxchi (hax being the contraption to hacking, browserhax = browser hacking), is not a coldboot exploit, you launch it yourself after booting the console.
cold boot hax chi (CBHC) is patching the console automatically at boot, you don't need to manually launch a channel yourself to get into CFW.
I'm not sure what the "chi" means in haxchi. maybe channel? or maybe it was related to the NDS game used by the exploit method? no idea. only Fix94 knows why he chose this name
On each console, lot of exploit and homebrew names are based after the console's name or its old code name or CPU/GPU codename themself.
for example, on Wii, the chipset are nicknamed "Hollywood" and "broadway" (officially), hackers unofficially nicknamed additional cpu functions "Starlet". all released homebrew had a tend to have two i in their names to look like Wii (ftpii, etc.)
The WiiU project name is "CAFE" (french for coffee), homebrew had names in relation to café :latte, Mocha, cafeiine (yeah, lot of homebrew kept the double i as it's still compatible with the wii), and the "iine" became another trend too for example with "loadiine", "emuliine", "injectiine".
Gamecube project name was "dolphin", that's why the emulator uses this name.
3DS installable file package name is "cia", hackers decided to call the CIA file format installer "FBI" to make it funny, etc.
same for the Switch, the console code name was "NX", and therefore lot of homebrew or hacks are using "NX" instead of "switch". it's operating system is called Horizon, the exploit launcher is named "fusée" (french for Launcher/rocket, rocket to launch a payload beyond the horizon), and CFW followed the trend and kept it all about space/spaceship.
It has a very weird "french word" trend too. Fusée, atmosphère, etc., now a little Mythology too (hekate, and other homebrew names)
not all, but most names have an origin, meaning and an unsaid trend to keep them all about a specific subject. it's more for fun than a real serious naming convention.
As for PSP/Vita, I always thought "hen" in henkaku or other exploit name was for
Homebrew
EN(abler). HEN is a temporary state of the console where you can access and launch homebrew, like haxchi for the WiiU it's not a persistent exploit and you have to launch the hack every time to "enable" it again, as opposed to full CFW state for the PSP with replaced files on the internal memory (M33, GEN, TZS, Pro etc.)
It probably became "henkaku" later for vita using the "hen" acronym with a name with a meaning. Maybe I'm wrong for Henkaku, I didn't follow Vita scene at all. but I'm pretty sure for "hen" being homebrew enabler.
edit:
maybe HEN is only a user-level exploit? you can launch homebrew, but no game backup/unsigned app? it enables only homebrew?
