I,m not sure why you care about the differences, but I grow both climbing Cecile Brunner and r banksia lutea (Yellow Lady Banks) right next to each other on the 3 story tall wall of my house. Since you are in zone 9, you should be able to grow both of them too.

Your bush does not look like a banksia rose to me - it certainly could be an old climbing Cecile Brunner. Have you seen the blooms? Are the small and light pink and have the form of tiny hybrid tea blossoms?

See the picture below - both of the roses on my house are climbers, and both can be trained (they don't have to get this big - I encourage the large size, because I love huge roses). In the picture most of what you see is Cl Cecile Brunner (pink blooms) - the yellow blooms on the left are from the edge of the huge yellow banksia rose. They both get the same height on our house (to the roof). They are both (only one plant) about 3 1/2 stories high and 15 or more feet wide.

The main other differences I can think of are:

1) the banksia is definitely once blooming, and starts blooming earlier (ours is coming into bloom right now). The CB usually blooms after the banksia has finished, which makes them great companions (this pic was taken because for some weather reason one year they were both blooming at the same time). The CB has scattered repeat all year, and a flush of bloom (way less than Spring) in the Fall.

2) the banksia is COMPLETELY, truly, thornless. The CB has plenty of thorns.

3) The banksia canes are much more flexible and longer than the CB canes.

Both can be pruned as much or little as you like. We only prune ours to keep the windows of the house clear, and to take off some weight in the Fall so they don't get torn off the house in Winter storms.

If I had your old CB I would probably cut off the top mess, clean the base, weed all around it, feed it, mulch it, and wait for it to put out more new canes, which could then be trained horizontally along the fence.

Jackie

Jackie, I really appreciate your thoughts! This is a new rose bush to me and I'd love to take a couple years and get it into shape.
I definitely have one CB with pale pink flowers that is doing well. This one has LOTS of thorns, so I guess it's not a Lady Banks. But it's in such poor shape I've only seen a couple flowers in the year I've lived here. They seemed mostly white to me, with maybe on a slight hint of pink.
Your home and roses are just beautiful!