Thanks for the suggestions. I don't think I've come across the
blackberry and tree killer before. I'll keep an eye open for it. I
bought some crystals a few years back which if sprinkled onto the
stump of a cut-down tree will kill the roots and stop it growing again
- that seemed to work quite well. Would this be effective on ivy if I
could sprinkle some crystals onto the cut stems just where they come
out of the ground?

I did some further searching and found comments that the waxy coating
of ivy leaves repels systemic weedkillers, as mentioned here too. But
it can help to crush the leaves underfoot or "with a spade" which
apparently allows the weedkiller to get into the leaves.

My guess is that this is not a different ivy but just the explosion of
growth of the original ivy, lighter green because it is growing fast to
cover ground.

I'm surprised to hear that. I think it is a different type. We have
had only the tiny three-forked dark green leaf type for the last 17
years and the new stuff is a quite new, although next door had it last
year, and it has run amok over the side of their house and outhouse.
The leaves are long eye-shaped and lighter green, and it stands up
from the surface and waves in the breeze.