Reality check: very few makers have “perfected” electric power steering. Here is a good summary of the current state of affairs:
Yeah, going from my '07 Civic all hydraulic to my '10 Insight all electric was a step down in feel, especially on-center.
Looks like true electric power assisted steering will be coming to the Miata in 2016:
In the specifications section of the Mazda website, for the MX-5 Miata, it says the steering is “Hydraulic Power Assist Steering (HPAS)”, which sounds like regular power steering to me. I also is completely different from the sticker you sent. You were clearly mislead by the sticker, which appears to be wrong.
Some reading leads to a conclusion that Mazda created a hybrid system that simply uses an electric motor to drive a conventional hydraulic pump, rather than some completely electric system, at least in some of its cars. You did say there is definitely a belt driving the pump. If there is, it’s not this hybrid system either. Personally, as a Miata owner for more than 10 years, I wouldn’t want anything else.
As to the Miata being twitchy, compared to most cars it is a little because of the highly responsive steering and suspension. Check your tire pressures carefully and experiment a bit, as the pressures really do change the way the car responds.
Yes, the Mazda website calls it “hydraulic power assist” but the Mazda sticker calls it “electronic power assist.”
IMO the most likely explanation is the system is conventional hydraulic with an electronically controlled valve that regulates the amount of boost, as described upthread. So it’s really a conventional engine-driven hydraulic system with an electronic control function added on top.
Probably the Mazda marketing department grabbed onto that “electronic” word and threw it onto the Monroney sticker to make it sound more advanced than it really is. Misleading, yes, but not an outright lie. Only a partial lie.
{ 
Caveat Emptor, as always.
Hybrid systems with an electric motor driving a hydraulic pump have been used for some years, mostly on German luxury cars, but even they are moving to simple electric assistance. I suspect that Miata uses that sort of power steering. Driving the hydraulic pump electrically is a bit more efficient than having it driven directly by the engine. Straight electric assistance is improving all the time, with the best being quite good. In a few years there won’t be anything else and it will work great. Just takes a little time to get the bugs out.
No, the OP has already said his Miata has an engine-driven hydraulic pump. The pump is not driven by an electric motor.