We look at things a bit differently.
My way of thinking...well...let's look at a computer.
You are saying that a computer is its processor running an operating system and you cannot really differentiate between the two. (this was a question)
Whereas I see that there is a processor that can run this or that operating system (Christian, Muslim, Atheist) which is one thing and that the processor actually running an operating system is another thing.
It is a question of perpective and also where the analogy break. At the most VERY basic level, you do not need a monitor, you do not need RAM you do not need a HD to have computer. You only need a processor with inbuilt RAM, I have programmed a few of those, where beside the board holding them and the input power there was nothing. You are speaking of a specially made expanded computer where the element are separated and bigger and intercommunicate. That is why the analogy break down.
But the basic fact, is that even in those big computer, the ONLY part calculating is the processor. The rest is only temporary memory (HD/RAM/Cache) output devices or input devices. Where does the program run ? In the processor only. The processor might from time to time get something from all those device, but it fully and ONLY run in the processor.
So taking the human analogy, when we say "my computer is running microsoft word" we are wrong. The "YOU" in this case being Word, and the "body" being the fully computer, the "brain" being the processor, it is painfully obvious that the YOU (program) is fully located and running in the processor and only there. The rest is only/input /output from the brain. The computer program is only running in the braina nd the YOu is only the brain.
IN FACT, if you were powering up and down the processor line manually without HD , keyboard or anything whatsoever, the processor would see no difference. I would wager it would eb the same for a brain : repalce allnever input by a simulated input for a brain in a jar , including details like heartbeat , and the brain inside would see no difference. That would still be a "you". Just like for that single processor with memory on board I used to program for my university degree.
PS: the reason I think the computer analogy with a processor is dumb and break down, is that today comptuer have RAM, HD, and processor separated (and heck even processor have ALU FPU separated on dye for example) whereas what we call the brain is a full set.