While I'm still less than a year old at cycling, my up-bringing has taught me to be courteous and considerate of others. Since I am a relative newB at cycling, and I don't want to give cyclists a bad reputation because of any ignorance I may have, I've been hyper-diligent at being courteous, and about following the rules and culture, just to be on the safe side.
With that said, I believe the "on the left" communication is more about how people interact with each other, whether they are on a bike or not.

While on a bike, I have observed that "on your left" is understood by cyclists to mean, there is someone on a bike coming from behind, and is going to pass on the left. Not taken as a command to get out of the way. If you are passing someone, you have to go out of your way to pass them. Not the other way around. Similarly, slower traffic is to stay to the right (perhaps this is a Texas thing) The left-hand lane is for passing, the same thought process applies to cyclists. (Generally speaking)

However I have noticed that non-cyclists (pedestrians) typically aren't aware of this term. From what I have observed, a phrase that is better received and understood by a pedestrian would be something along the lines of: "excuse me", "pardon me", "bicycle on your left" or any other term that one pedestrian would say to another. AND, you don't whiz past them, shout, or do anything that would startle them. Basically, be nice.

Cyclists are thinking about other cyclists being on the road/trail. Pedestrians are thinking of (expecting) other pedestrians being on the road/trail. Generally speaking.
So if one is to pass another from behind, one should be considerate and courteous.
Perhaps this doesn't apply all across the world, but that's how we do things in Texas. And this is how Texans should behave anywhere they go.

These are just my observations.