The present perfect conceptualizes the past thing from the point of view of its relation to the present.
Having eaten pumpkin pie, I know what it tastes like.
My present knowledge is the result of eating pumpkin pie at some point prior to "now". The eating might have taken place 1 second ago or 1 year ago or 10 years ago. I might have some unfinished pie on a plate in front of me. All we know from the tense is that I am someone who ate some pumpkin pie at some time in the past.
With "having lived", if you said this and only this:
Having lived in India, I am familiar with the cuisine.
all that can be concluded from the statement is that you lived in India at some period in the past. It could be last week, or last year, or for the past 20 years. You are a person who can truthfully say "I lived in India". You might live there still.
Now, let's see if we complicate things by adding "all my life".
Having lived in India all my life, I am familiar with the cuisine.
All my life does not exclude the present moment; nor does it necessarily include it. You can easily say, as someone who has emigrated to Antarctica, "I lived in India all my life. But now I live here at the South Pole. Aren't the freezing temperatures refreshing?"
So, we cannot conclude from the statement whether you are living in India now, or have moved elsewhere. All we know is that you are a person who can claim to be someone who, at some time in the (recent) past, had been living in India since birth. all my life does not exclude the present but it does guarantee it: a person who had been born in India and who had lived there until very recently, could say "Having lived in India all my life..."
So, yes, you can say "having lived" whether you have moved away, or you still live there.
Best Answer
In American English there's a subtle difference:
If the Male Passenger had said "She got her bag caught" the Subway Policeman might infer that the Male Passenger had witnessed it getting caught, whereas "She had her bag caught" is simply a way of referring to the existing state of affairs that explains the Male Passenger's involvement.
"She had her bag caught" could be paraphrased "Her bag was caught" whereas "She got her bag caught" could be paraphrased "Her bag got caught".
Compare: