There is no specific name for spinal fusion surgery.
Even though there are a variety of techniques using different substances (such as recombinant human bone versus bone taken from another part of the body), names have not been developed or used to differentiate the types of spinal surgery being done.
Also, whether the surgery is a first step or is following an unsuccessful titanium cage insertion, the name is XXXXX XXXXX fusion.
The only "classification" refers to how the surgeon is going to approach the spine.
In posterolateral gutter fusion, the surgeon opens the back to get to the spine. He/she does not actually remove the disc but does add bone to form a graft.
In posterior lumbar interbody fusion (PLIF), the surgeon opens the back and actually removes the disc located between two vertebrae. He/she then inserts the bone grafts into the space.
Anterior lumbar interbody fusion (ALIF) is the procedure when the surgeon approaches from the front of the patient. Here, too, the surgeon removes the actual disc and replaces it with the bone grafts.
Finally, anterior/posterior spinal fusion refers to the procedure when it is done with an approach from the both the front and the back.
There really are no particular considerations for the procedure that apply to everyone. Your surgeon needs to evaluate you, all of your imaging (MRIs, CT Scans) and decide how to best approach your situation and what to do about the failed cage currently in place.