Is secure erase the equivalent of low level format or is it something deeper?
Deeper. Secure erase writes zeroes to every data space on the card, either 1 (not secure), 3 (better), or 7 (CIA secure) times. Low-level format writes over the data space once, and the File Allocation Table once. For most of us, low level format is quite sufficient. Jeez, for me, ERASE ALL is good enough. Nothing I do is all that important.
I use Recuva to retrieve accidentally erased files, not erase files.