As an academic, and non-native speaker of English, the issue strikes fairly close to home. But the exact nature of my discomfort is somewhat indirect.
Let me explain.
I came to the United States almost exactly 11 years ago. Like almost everyone else, I was initially amused by differences in American spellings ("color" not "colour","check" not "cheque") , pronunciations ("zee" not "zed", "semi"), and syntax ("meet with me"). However, over a period of time, I internalized this "fork" of standard English, like almost everybody else around me.
Indeed, many Americanisms felt more natural, and less contrived ("meter" over "metre").
Over the last decade, I have also noticed that my speech indeed sounds different, depending on the environment I find myself in. The amount of "Indian" or "American" accent that gets dialed in, is variable - even at the risk of sounding inauthentic or phony.
Recently, I visited my 7 year old niece, who has lived in England for the past five years. During the first 2-3 days, I was amazed that she had not yet fully picked up a British accent, since most immigrant kids of that age in the US usually do.
Then, one day we visited one of her friends, and boy, did she sound different! She spoke without the slightest trace of an Indian accent.
Even as a kid, she was calibrating her speech to her audience!
Whether the motivation was efficient communication, or not sounding like an outsider, I still don't know.