compiler can be the common arm one
see here: https://launchpad.net/gcc-arm-embedded
that's sort've the gold standard open compiler for all ARM "microcontrollers", meaning, ARM Cortex-M series chips
and I think "arm-none-eabi" can be just understood as "arm compiler for Cortex-M" without fully grokking each word in the phrase
eabi is just the universal binary ABI for ASM calls this compiler adheres to
what you need from TI is TivaWare, which has the headers and C source files for the peripheral drivers.
along with a lot of other cruft you won't care about (in part b/c it's not even open source)
here's ARM's page on Cortex-M: http://www.arm.com/products/processors/cortex-m/
that'll give you a lot of marketing BS but general information about what arm's "microcontroller" offerings are about
where TI, Freescale, STM, etc come into the picture is they "license" the ARM cores, add their own peripherals on top and produce a final chip that's tailored for whatever use cases they feel are profitable for them to target
I was told once by a TI'er, for what it's worth, that the TM4C129 (Tiva-C w/ Ethernet) series chips were initially conceived of in TI as a solution for baseboard management controllers in high-end servers
so you have to look at the whole chip and its peripheral set from that light.... this is what they "had in mind" when they designed it. You almost never get that kind of information from a vendor though.