sorority letters of support

Unless these are UT alumnae speaking from experience, I would get the letters.

They probably don't want to write a letter of support because they don't know you. Those are needed but usually written by an alum that knows you well and can vouch for you.

Houston Panhellenic gave same instructions for Kappa, KD and AXO

That must be coming from their main offices. However, it's delusional to go through recruitment in Texas/the South without recs. Get them from someone who knows you/your family well.

There is a difference in terminology between chapters, so that may be some of the confusion. I would get references/recs for each chapter (whatever that chapters official recommendation form is called). There are also some general letters of support that are just letters (not on the official form) that some chapters either encourage or require. Those are just personal letters from an alumna on your behalf.

I would say recommendations/references (the official ones for each chapter) are the most important, though some chapters may like or require those other letters.

I am an alumn from one of the chapters you mentioned, and the information you have been given is correct. Some chapters are only requiring recs and not letters of support. This is the case across the nation, and not just at Texas. Plano Panhellenic will have the latest information on what the specific sororities are asking for.