People seem to refer to the bond yields rather than prices - which are inversely proportional to bond price, and hence proportional to interest rates.
The way I understand it, is that as you say if interest rates rise then eg a £100 bond with a £4/year coupon becomes less attractive, so the price of buying that historic bond falls. So perhaps you can buy that £100 bond for £80, which means that you're getting a £4 coupon for an £80 bond, ie yield has gone up from 4% to 5% And of course, the government has to reflect that in newly issued bonds: so they will sell a new £100 bond with a £5 coupon.
So bond yields (which that graph shows) should go up proportionally to interest rates - and it's any discrepancy from that being proportional that is due to distrust in the government.
Interesting to compare German and UK bond rates! That does seem to show a very similar rise as interest rates went up. And seemingly only a brief jump for Truss, (yellow highlight) which seemed to come back into line if you compare the shape to the German bonds.
We're a full 1.5% higher to Germany. But oresumably that actually reflects German inflation being only 2.1% currently compared to our 3.4%.
So presumably, the 'idiot premium' is just about how inflationary the markets think a government's policy will be?