How do I break a concrete slab without damaging cabling running through it?

The other answers are good. If you do decide to go ahead with hand tools, I recommend using a digging bar (it has many names).

Use it vertically, hitting the concrete with the blunt end, causing the concrete to break at its weakest places. Fewer chips flying around, and safer.

https://en.wikipedia.org/wiki/Digging_bar

If you need to maintain the existing phone cable, leave it alone. Actually, even if you're not planning on using the existing phone cable, just leave it alone.

If you want/need to run the fiber close to the existing cable, drill a new hole. That will give you a neat, tidy opening you can run the fiber through and shouldn't cause any damage to the existing cable, so long as you're not trying to get too close to it.

Optionally, run the fiber in a totally different location. Bring it to the slab, then run it up, in EMT or Schedule 80 PVC (for protection on the outside of the house), until you hit wood, then go in through the wood instead of through concrete. This will make all your work much easier as any work to get through wood is much easier than any method of getting through concrete.

I understand the desire to do it properly and fix the issue, but cutting concrete is a lot of work and is probably going to rip your wire apart in any case, no matter how carefully you try to break it. So isn't it easier to lay new wire behind/above/around the concrete slab, or if available, using a different tube for electric wires that is also already cast in there? Tidy up the old wire and cut it off at both sides of the slab, but otherwise don't touch the concrete?

I would probably want to cut the slab, using a masonry blade on a skill saw and leave that small chunk intact it is really hard to keep from damaging wires through concrete the vibrations tear up the insulation. Once you get the slab removed you could try to break the small piece up but be ready to splice in a new piece using a wet location communications crimps. The ones I have use a dielectric jell you slip the 2 wires in and smash with pliers.