Aside from the tire setup you have with 245 in front (what's in back?), I also double what @engifineer said about caster and to check yours, if you've never played around with it. I run mine with more caster and like that 'heavier' feeling, and it results in what appears to be a harder turning radius at lower speeds but is generally a more heavy or solid steering wheel than with less caster. I didn't like less caster as the car would practically turn sideways with the smallest of touches to the steering wheel.

I like more caster as it gives more (perceived maybe not sure) stability at high speeds and on long sweeps. Lower caster was great for zipping around cones in a parking lot at 15 mph but as a daily driver for long drives and when doing higher speeds the car was too twitchy and a handful (for me). Going triple digits with low caster and I was white knuckling the steering wheel, more caster and the car isn't picking up every single tiny line in the road, more confidence to push the car harder.

It's amazing how much the caster impacts the the handling on this car.

 

Sounds like you need to attend an S2000 meet, to compare. You may have no problem at all.

 

I think this was the right test. Stabbing the wheel left, it returns to center. If anything it over-corrects. Stabbing the wheel right, it does not return to center. (Turns from lock-to-lock are the same. About 1.25 turns each way.)

I'm going to try re-greasing the EPS torque sensor. Especially after the ***unrelated*** dramatic difference it made regreasing my clutch release fork. I had a 1-2 grind, and now it's gone! It was pretty nasty and obviously needed attention once I saw it. I assume the torque sensor is similar.

 

The thing that needs regressing on the torque sensor is the slip ring. It only moves like 5mm, if it doesn't move smoothly you will have inconsistent power steering.

I bought a replacement torque sensor with a broken housing, but the torsion bar with the slip ring was in good shape. I transferred it over to the housing on the original and boom, works perfect.

If the grease dries up, the pegs on the torsion bar will wear grooves into the plastic slip ring. Once you get there you have to replace the torsion bar.

You can source the torsion bar from a Mazda 6 made from 2006-2012. It's the same bar and sensor, even though the Mazda uses hydraulic power steering. The RX8 one won't work, and the Honda torque sensor is like $400.

 

Update: re-greasing the EPS torque sensor helped a little, but did not fix my issue. Replacing the EPS sensor with a new AP2 unit fixed everything. Now my power steering finally works correctly!!

As confirmed by several other threads, the AP2 torque sensor is a direct replacement into the AP1. I was initially a little confused because they're different part numbers and a reddit post said "it's no longer available from Honda". But the AP2 unit fits and functions exactly the same.