[talanum1 (OP)]

Light cones start at a point in 2-dimensional space. I am a observer and not a point so how is my light cone defined? Also I see persons next to me and can remember their position in the past. Do the origins of  our light cones overlap?

[Origin]

Light cones start at a point in 2-dimensional space. I am a observer and not a point so how is my light cone defined?

After 10 seconds your light cone will have a diameter of 6 million km, so it is a good approximation to consider yourself a point. 
 

Also I see persons next to me and can remember their position in the past. Do the origin of  our light cones overlap?

Let's assume you and the other person are 100 km apart.  The origin of your 2 light cones do not start in the same place but they are very close.  After 10 seconds your light cones will essentially be the same.  That is, both light cones will have a diameter of 6,000,000 km but the will be offset from each other by 100 km, which is negligible.