Straightness on a circle in the simplest form, means that the whole length of the horse from nose to tail is on the arc of the circle.
However, if you picture it, and imagine railway tracks going in a circle, the outside track has to be longer - outside edge of the horse is travelling on a longer track, the inside of the horse on the shorter track, but more acutely angled.
If the horse is travelling straight along these tracks round the circle, its shoulders lie on one radius of the circle and its hind quarters on another radius.
But the two lines out from the centre are not parallel to each other, so the horse's shoulders are not parallel to its hips.
The great, enormous, unsettled question is how the rider should be sitting. Some people say that the riders hips should be in line with the horse's hips and the rider's shoulders paralell to the horse's shoulders. But since the shoulders and hips of the horse are not parallel to each other this involves a twist in the rider's body - sometimes referred to as the German twist.
For most of his life Charles Harris advocated that the rider too should sit on the radius of the circle. Not twisted at the waist. Not parallel to the horse, but on their own radius. This is what I was taught to do. To line up my shoulders and my hips with the centre of any circle, starting with the RI holding the lunge line. I find this useful; out hacking it taught me to canter quite fast round bends in the track and keep my balance. It keeps the rider straight and secure on a bending horse and corrects the tendency to put your outside shoulder forward, thus releasing the outside rein.
So I would say that straightness on a circle is best understood as straightness in relation to the centre of that circle.
This remains the situation even if you arre asked to ride a circle with counter bend.
Going just a little further, Mary Poppins, as one leg yields out from the centre, the shoulders of the horse, the shoulders of the rider and the hind quarters of the horse, all move outwards along their three radii from the centre of the circle.
And the next part of the exercise is usually to spiral inwards towards the centre again, inclreasing the bend (reducing the circumference of the circle, but still both of you remaining straight in relation to the centre of the circle.
Now when you go straight - the two railway lnes of the inside and the outside of the horse - travel exactly the same distance. The outside legs of the horse do not have to travel farther than the inside ones.
I dont know how your RI wants you to ride corners - but one way of riding them is to imagine an arc of a circle - and to imagine the centre point of that circle too and to keep "straight" on it.