There is a difference between the Pilates description of pulling the bellybutton to the spine, which works well especially when lying down, and the ballet usage of your lower abs. It is indeed in the very low and deep abdominals, and if you pull those up you’ll see your bellybutton lift up. That movement pulls the abs up and flat, like a supportive wall. It doesn’t bunch the muscles up.
These deep abs, along with your turnout (rotator) muscles, your thighs and some support by the gluteals, stabilizes the core area.
The gluteals (butt) muscles do not need to be clenched, but will automatically activate when you turnout.
The back will tense in response to sucking in/up those abs, as the muscles work together. However, you don’t need to focus on maintaining that hold, focus on the abs. Tension and release is fluid, and that is part of the control gained by years of ballet training. Ribs and shoulder areas are always moving slightly. If the core is strong, the upper torso can be balanced and stable, but you can still breathe properly and allow easy head movements and easy back bends. By that I mean, the same, a strong core area that allows the back bend and recovery without strain.
The one exception to the fluidity of tension is the deep low abs, your support that holds strong. This abdominal support allows the lower ribs to expand for effective breathing. Done correctly, this actually helps fluidity as opposed to weakening the tight hold, which should be in the low abdomen.
Hopefully this fluidity flows down to your finger tips. Your thumb muscles like to take on a lot of tension, and many dancers work with thumbs sticking up to some degree, or clenched into their palm.
Also, a held spiky finger formation, especially an index finger sticking up, shows a chronic strain and lack of core control. It is also an affectation that can be copied from a teacher or another dancer. Energy running through the hand, slightly straightening the palm and fingers, is different than tension. Of course the tension changes during multiple pirouettes and tours en l’air. The virtuoso tricks of ballet call for a tightly held ballet position, yet the additional tension can be released immediately upon reaching a landing pose.
Whether you are a would-be ballerina, among the men in ballet, or a recreational ballet dancer, core control will help you achieve the elegance and beauty of ballet movements, and other dance styles too.