Instructor is most important, and I echo and endorse all of those sentiments.
That said: I have an IANTD technical cave card, and I have never had an issue with diving caves on it. I have had issues getting O2 fills with it.
IANTD has too many levels of certification which include too many things, while TDI is much simpler. My one IANTD card covers what TDI calls "advanced nitrox, Decompression procedures, cavern, intro to cave, full cave, and extended range." (Asterisk on extended range: tdi is 180ft, while iantd is 170ft). However, there are other iantd cave cards that do not cover all of this.
Why is this important? Because I used to have to sit there and explain what my card covers, and wait for them to call their agency for equivalence. In the end, I went back to my cave instructor and asked for an advanced nitrox card so I could get O2 fills. I still sometimes have trouble with boats saying I am only trained to do deco in caves (not true).
TDI makes life a little easier. Why? Want to do an open water deco dive off a boat?... whip out your ANDP cards. Want to pay cave price at Ginnie...whip out the full cave card. Everyone knows what they mean.
Also, TDI's elearning for Decompression Procedures is top notch (cannot speak for all the others).