Add new lighting before previous lighting doesn't work
"It works" is not a measure of quality.  The first problem I see is that someone has confused black and white coming from the power source and they go the opposite places from where they should!  Exchange them. 


Power supplies deliver Always-Hot and Neutral. 
Switches require Always-Hot and deliver Switched-Hot. 
Lamps require Switched-Hot and Neutral. 


If you are wondering "What is the difference between hot and neutral? This is AC Power" that's beyond the scope of this answer, but it has a real answer.  

Now you notice all cables come in black, white and safety ground. Yet we already have named 3 wire functions - how do we get 3 wire functions on 2 colors of wire?  If any random wire can have any random function, this rapidly becomes exactly the dizzying nightmare you are dealing with. Anyone would lose their mind!  The way to regain it is something not required by Code (much): 



Now let's see what our wiring looks like.   Now, white = Neutral, black = Always-Hot, red = Switched-Hot.  We will tape wires as-needed.  Wires marked with tape are now that color.  



This is legal/correct.  I fixed the crisscrossed supply wires.   Note how white is remarked black when there's already a black there, that satisfies two Code requirements.  (Actually, since 2008, Code requires real neutral also be supplied in switch loops.) 

Marking the Switched-Hot wires red is not Code mandatory, but boy, does it make this diagram clearer! 

Now let's do that to your second diagram and see if it gives us any clues. 



Well there's your trouble!  Lamps don't take Hot and Switched-Hot!  This can't work. Lamp needs neutral, and it has to go get it from somewhere.  

Perhaps it's time to modernize the switch-loop wiring to the /3 cable which provides real neutral also. E.G.