Exactly.
I thought that's been understood for decades. The LA times discussing that from the 1990s, probably the 1980s. West Hollywood and Beverly Hills have always been part of West LA/westside of LA, whatever you want to call it. Otherwise, what else is it? Central LA? Most people wouldn't agree to that.
Even if you think the westside/West LA starts at the 405 (some do) how many Angelenos talk about West LA and refer to that small neighborhood near Brentwood?