Don't feel bad. The reason you're getting so many suggestions is we've all had troubles with bilge pump wiring
All of the suggestion for connections are great but I would first get it working with temporary connections. Then do a permanent job. There's a chance the thing is just broken out of the box.
If the old system EVER worked, you do not need to "rewire the boat". Christian's diagram is the one. But to repeat.
The black wire "coming into the bilge" should go to the pump AND the black of the water
The red "coming into the bilge" should go to the red of the water
The brown of the waterwitch should go to the red or brown of the pump.
While soldering is fine here, crimped connections are often better in a boat because the solder can crack from vibration and movement. Making a good solder connection is also not easy for the inexperienced. If you do decide to use crimped connectors, get a good crimper like the Ancor Double Crimp Ratchet Tool - costs less than $100 and will last a lifetime.
Bilge pump wiring
is also one of the most common ways that stray current
gets into the bilge
and cause galvanic corrosion
. No matter how well sealed, keep as much bilge
pump wiring out of bilge water as possible. If any wire must be normally wet, check it carefully for any cracks in the insulation
The Ultra is from USS who also make Tef Gel for rigging
TEF-GEL - Ultra safety systems - Home page