@ed_crumbpacker: Welcome to the forum!
ASSUMING things are still as they were back in the day...
CORE is the kernel, busybox and just the very basics/necessities - you then build onto TCL to suit your needs
TINYCORE was CORE plus a very "Tiny" (size/memory) desktop environment, but not too much of anything else -
you build it!
PICORE is usually a TINYCORE-like environment, just intended for ARM based devices like Raspberry Pi
dCORE from my understanding is TINYCORE, but based off of
Debian extensions.
MicroCore - I don't think that name has actually been used since version 2.x or 3.x but is actually on the home page if you scroll ALL the way down under "News" to 4.something or later I believe. Think of my "CORE" definition above as today's MicroCore.
CORE and TINYCORE should be able to run on most 32 and 64 bit machines.
Anything "PURE" or specifically has "64" in its name is usually 64 bit only
(it boots on my dell 32 machine but not on my 64 toshiba's)
I'm guessing your Toshiba probably has UEFI (Secure Boot) turned on; if you disable UEFI you will likely be able to fire up TCL.