Only the second sentence is correct.
The period 2001-2009 is a finished period of time. She no longer lives in London, so you have to use a past tense : 'She lived in London from 2001 to 2009.'
There are two main situations in which you would say 'She has lived in London'.
1. If she still lives there now and you want to say how long she has lived there. In this situation, you would say 'She has lived in London since 2001.' or 'She has lived in London for fifteen years'.
2. If you are talking about her life experiences in general without a time specific time period: 'She has travelled a great deal. She has lived in London. She has also lived in several other European cities, including Paris and Berlin.'
I hope that makes sense. Please ask if anything's not clear.