In my younger days I designed mine haulages and winders for a living. I knew a bit about wire ropes for those purposes.
Though I never had to design a capstan winch, the design handbooks give the following wire rope requirements for capstan winches:
- Ordinary lay - the most common lay, where the wires of the strands are laid in one direction and the strands are laid into the rope in the opposite direction (this helps prevent the rope twisting as it stretches).
- Should be flexible - flexibility increase as more, smaller diameter wires are used to make the strands.
- Should provide a good gripping surface on the face of the capstan drum.
- Should be reasonably resistant to wear. Small diameter wires don't wear as well as larger wires.
Recommended construction for capstan winches is 6x24(15/9/F)/F. This construction has 6 stands with 24 wires in each strand - each strand has 15 outer wires over 9 inner wires laid over a fibre rope core, the strands are laid over a fibre rope core.
In 9mm diameter, the outer wires would be approximately 0.48 mm diameter. The weight would be approximately 25.8 kg/100 m. The minimum breaking strength is 35.6 kN (3629 kg, or 8000 lbs force) for 1570 MPa wire.
This is obviously not the construction that you have because yours has a wire rope core.
The closest rope construction to that but with wire rope core that I can find listed in 9 mm diameter is 6x36(14/7+7/7/1)/IWRC. In 9 mm diameter, the outer wires would be approximately 0.50 mm diameter. The weight would be approximately 33.9 kg/100 m. The minimum breaking strength is 51.1 kN (5209 kg, or 11,484 lbs force) for 1770 MPa wire.
When a wire rope is bent over a drum or sheave, the stress increases in the outer wires of the rope - this is possibly (but could have been previous damage at that point) why your rope broke at the capstan drum (where the wire stress is highest).
The ratio of the drum (or sheave) diameter to rope diameter and the ratio of the drum (or sheave) diameter to outer wire diameter is an important factor to reduce the stress rise in the wire to an acceptable limit. As drum (or sheave) diameter is increased, the rope and outer wire diameter can be increased.
The rope diameter (9 mm in your case) is nominal. The diameter of a new rope will be larger by as much as 7% for a 9 mm rope (approximately 9.63 mm).
The swaged terminal on your rope is similar to those used for yacht rigging. A good wire rope merchant should have these and the tools to swage them, or else try some of the places that do yacht rigging. Perhaps you can find a Ronstan catalogue on line.
Edit: BTW, 6x36 IWRC is commonly used hoist rope for cranes.
Edit 2: My very old Ronstan catalogue has a swage eye for 3/8" wire that might do. It lists 8 and 10 mm but not 9 mm, but newer catalogues might have 9mm. 3/8" is 9.52 mm. The part number for 3/8" swage eye is RF1500-12, if they had a 9 mm swage eye it would look like RF1501-9.

