Arduino: ARM, MCU, SoC and RTOS
My understanding is that ARM is the chip architecture (a conceptual design, really),


ARM is the CPU core architecture used in the Atmel SAM3X MCU family. The MCU is the CPU plus all of the facilities (PWM, ADC, DAC, DMA, etc.) that are implemented in order to allow the CPU to interact with external circuitry.


  and that Arduino simply implements this design,


Most appropriately, "Arduino" is the IDE and the software libraries that come along with it, plus the specifications and connector layout of the various pieces of Arduino hardware.


  and is also a "system on a chip" because it has other things (USB ports, GPIO pins, etc.) that make it more of a computer than just a MCU.


"SoC" is exactly what it says on the tin: it's an entire system on a single chip. The Due is not a SoC, since it's a full board. The BCM2835 found on the RPi is a SoC. The SAM3X is a MCU, since it still requires more than just connectors to become a full "system".


  My understanding is that if embox runs on ARM, which they claim it does, then at some point in the future I can take my application and run it on another ARM-based system without any real headache/pain.


Provided that embox runs on the new system and that your application can be compiled for it.