I’ve seen lots of different descriptions of bracing the core (suck in/hollow out belly, contract abs as if getting ready for a punch, etc.). Is it primarily tensing the transverse abdominis (TVA) or the TVA and the lower rectus abdominis (LRA). If you tense the LRA, don’t you move the pelvis into more of a posterior tilt and move away from a “neutral” spine?
A perfect example of where you can easily do a movement both ways is the supine bridge. If you contract the LRA, you’ll feel you are rolling up the lower vertebrae as your hips lift. If you use only the TVA, you will maintain lordosis as you squeeze the glutes to lift your hips.
It would seem that using the LRA is good to improve lumbar mobility (especially those with anterior pelvic tilt and weak LRA), but for resistance training for the legs, it seems that you would want to maintain a neutral spine and pelvic stability - thus, tense the TVA, but not the LRA.