API gateway is a fancy term for a configurable reverse proxy often bought as a service.

No, API gateway is a web service that has a main purpose of routing requests based on incoming requests.

Load Balancer maing purpose is to...balance the load across multiple backends.

Just because both can be implemented with a reverse proxy such as NGINX doesn't mean it's the same thing.

I said "API gateway is a fancy term for a configurable reverse proxy often bought as a service" and both "load balancer" and "API gateway" are common configurations of "configurable reverse proxy", often bought as a service.

Many load balancers have this functionality within them, even ones from years ago that aren't around anymore like Microsoft ISA/TMG. They're not web services, but they can route based on requests.