Is there any difference between "putting on" and "faking" (an accent)?

I would like to suggest a slight difference in meaning and then illustrate it with a personal account. I start with a choice (from many) of two meanings:

To put on = to appear to have a feeling or way of behaving that is not real or not natural for you

Cambridge dictionary

To fake = : Counterfeit,Simulate, Concoct

“faked a heart attack”

Merriam Webster

Considering these definitions, to fake is to simulate, whereas to put on may be to adopt something that is not simulated, may even be real, but is not natural to you.

My example is that I am English but have lived long enough in Scotland to be able to put on a West Scottish accent and word choice sufficiently good to occasionally persuade a Glasgow taxi driver that I am local. When I do this, my language is not faked, because I have learned it well; but it is put on, because it is not natural to me.