That doesn't sound right either. Bit I suppose that when the button is pushed, the light coming on is just the indicator of other things coming on, such as computers, sensors etc.

In my plain-jane  2010 Suzuki daily driver, when you turn the key (no button!) to the on position, before cranking the engine, all the instrument lights come on as a test and the fuel pump is energized. Even my 1980 Mini does that.

I live in fear of having to buoy a modern car with the keyless fob, push button, lane assist, automatic braking and all that crap I can do myself.

OEM radio. For the cars I’ve had in the past the only way to get the stereo to function is to have the key in and it turned to accessories. It’s a bit strange that the Mini stereo functions with the key out. I thought that 2 amp draw was a lot until the start stop button light came on and the amp reading just about doubled??

Modern car "radios" have two power feeds - one to actually play it, which might be around 2 amps (dependingdisconnected,  on amplifiers etc), and the other, a much smaller load to just keep the memory pre-sets alive. That should be much less than 2 amps. That is why the settings are lost )among other things) when the car battery is disconnected.

The newer in-car display screen that include all sorts of touch-screen controls, GPS are a whole other ball game. But they are all computer managed, so can be hopelessly complicated.

If you are talking about something more conventional (stereo, radio receiver, CD player etc.) then it should poser off when the car is off and the key removed. 2 amps is a big draw.

Is it a MINI OEM system or something after-market?

It might be easier if I just ask someone with the same model Mini if they have power to the radio when the car is turned off and the key removed from the ignition. 

I don't know if your method of reading amps is part of the problem. Instead, try measuring voltage between points of the circuit and ground. There should be a slight difference in voltage from one point to the next and a bigger drop before and after the load (radio). If the Voltage is near zero from a point to ground, you might have narrowed down where a short may be. 

On Youtube, search for "south main auto parasitic draw" 

He is a wizard at finding electrical problems in all sorts of cars.

Trying to find parasitic current problem in a 2012 JCW. Drawing over 2amps when car is parked. Pulled fuses until I found the problem is the radio. Still power going to it when the vehicle is off and key is out? Also the start stop button randomly lights up when everything is off. Not sure if it’s coming on when I have the amp meter connected up when I’m fault finding. Anyone had similar problem?