I lived in L.A.
I have lived in L.A.
Both of these mean that you lived in LA in the past. The second version is not usually used as a stand along sentence.
You would say:
eg. I have lived in LA for two years.
eg. Q: where did you live before you lived in London?
A: I lived in LA
I live in L.A.
I am living in L.A.
I have been living in L.A.
These three all mean that you are currently living in LA.
There is a slight subtly between I live in, and I am living in. I live in is more permanent. I am living in implies that in the future you might move, or in the past you may have lived somewhere else. This is just because it highlights the current action on living right now.
eg: Q: Where do you live?
A: I live in LA.
eg. Q:Where are you living at the moment?
A: I am living in LA.
I have been living implies a period of time. It is where you currently are, but you are stressing a certain amount of time. this could be sort term as in
eg. Q: Where have you been living while you are studying your degree?
A: I have been living in LA.
Or the time period could be long.
eg. I have been living in LA all of my life.
I stayed in L.A.
I have stayed in L.A.
This follows the same rules as I lives in LA or I have lived in LA. It means you have stayed there in the past.
Here you are not saying that you lived there- but rather just staying.
Often these words are used interchangeably, but living means this is your home. It has more of a sense of permanency. Staying usually means that you were just there for a temporary amount of time.
eg. I was staying in LA while I was looking after my grandparents, but home is still in New York with my friends.
I am staying in L.A.
I have been staying in L.A.
I stay in L.A.
These three are again the present tense. And follow the same rules as
I am living, I have been living, I live in