It's funny how different people define "Westside" differently. Could be a generational thing.
I'm a Gen-Xer born in 1970; for me, LA's Westside was always west of Beverly Hills and north of Venice. So, Century City, Westwood, Bel Air, Sawtelle, West LA... those were all "Westside." Beverly Hills was always referred to as Beverly Hills. Same with West Hollywood. If you were in Beverly Hills, or going to Beverly Hills, you said you were going to Beverly Hills or "I'll be in Beverly Hills." You never said you were going to the Westside. Venice had/has a strong enough identity that it's also considered its own entity. I've heard people refer to Marina del Rey as the Westside, but that's iffy for me. I consider Westchester to be "LAX." It's funny... years ago, I had a co-worker that many of us couldn't stand. She was an odd duck... and she was a GEM (sarcasm, of course). And she would always say that she was from the "East Coast," but she wouldn't be specific as to where (I've encountered other transplants who were like that). I learned later that she was from somewhere in Pennsylvania. I'm like, 'why don't you just say where you're from??' Anyway, we had to attend a seminar at the Airport Hilton on Century Boulevard near LAX. I ended up having to sit next to her. Her cell phone rang, and it was her husband. She said "Didn't I tell you I had to go to a seminar?? I'm on the Westside! I won't be home for HOURS!" And I was thinking like 'Uh, this ain't the Westside, lady!'
And she lived in Sierra Madre.
South of LAX is the South Bay, and I include Inglewood as being part of the South Bay.