If you only started lessons two years ago, my worry would be that you are basing your love of forward horses on one horse. Maybe the two of you just click.
There are different kinds of forward horses and different kinds of "Steady Eddies" (school horses).
At some point, as skills progress, riders do "outgrow" certain horses. If you were only exposed to those "zombie schoolies" - those horses that are truly on auto-pilot despite you giving them any cues, I can understand that two years in, you might be getting bored.
And, yes - you might be a person who likes forward horses. But, imo, I think you might be just feeling the fact that the horse is actually listening to you - and it just happens to be a forward moving horse.
Here's my worry: With two years experience, if you start seeking out only forward horses, the odds are that you will get on one that is too much horse for you. Unfortunately, there are a lot of described "forward" horses out there that are "forward" because they have not been trained properly. That's when an accident will happen - and those of us over 40 don't bounce as well as we used to.
I agree it does sound like you have probably outgrown the basic schoolies....so, I would look for lessons on a more advanced, trained horse - a "schoolmaster", if you will. That way, you will continue to learn about how to become a better rider and, if you then decide you definitely like forward horses, your skills will be better honed to deal with anything that happens at speed.
I know you don't want to show or anything - I'm more concerned about your safety.