Hmmm... I have had many kidney stones, and I've not experienced the same type of pain with them that you describe. Kidney stone pain is typically "colicky." Meaning that it's almost like a rhythmic, squeezing pain typically in the flank area of your back. It can radiate down your side, into the abdomen and pelvis on the affected side, and it is often severe.
It also doesn't usually get better if you sit down. Many people with a kidney stone attack find it very difficult to sit still and often feel better walking around. It also usually hurts only on the affected side.
I'm not a medical professional, and I certainly can't say that it's NOT kidney stones, but mine usually present differently than what you've described.
I hope you find some answers and relief soon.
Hugs - Austen