But would you really call this concrete structure you see in the picture a river?

If the LA River is any historical precedent, yes.

If the LA River can be a river, why not?

The flow of water was there before the concrete constructions were made, I guess?

Not necessarily. It's common practice to built structure to control and redirect rivers around developments. That doesn't stop it from being a river.

I don't know the specifics of the LA river, but I assume it has been a river long before the supporting concrete structures have been built. If they exist, they probably have been built to support the river in a way to not let the land around it sink into swamp land. I think that's pretty different from constructing a concrete canal directing water, which hasn't before existed. Or so... I think there is some difference.