bare metal vs. RTOS vs Arduino

I've been googling around trying to understand what the difference is between those terms and trying to fit existing platforms into some kind of understanding.

I kind of understand the features of RTOS and the features of Arduino. But I don't really understand how RTOS and Arduino might be different than, say mbed. Or how that's different than using Silicon Lab's or ST Microelectronic's IDE's to make applications. I don't have boxes for putting these different things in.

Is there an easy way of understanding this?

If you programming controller without using any [RT]OS it's a bare metal approach. And it doesn't matter which IDE you are using.

Arduino is bare metal (meaning there is no OS) - the difference between using say, atmel studio vs arduino is that arduino does a few things to make life easier for you:
It provides convenience functions like digitalWrite/etc and an implementation of Serial, and libraries for things like i2c and spi, as well as compatibility eith all the arduino-compatible livraries availanle everywhere
It generates function prototypes for you when processing the .ino file (libraries still have to do this themselves)
It initializes peripherals before the sketch starts running (so that pwm and such will work)
It then calls setup() and repeatedly calls loop()