Unless there is a hose barb fitting on the end of the pipe, or at least some type of inverted flare, the rubber hose WILL slip off the pipe. The reason it cut the hose the first time is because they probably only did a single flare on the end of the pipe instead of a double flare. The single flare always leaves a sharp edge.
Most aftermarket tranny coolers will provide universal fittings that will go on the end of the factory cooler line and in the radiator tank, allowing you to safely connect a rubber hose. One of the fittings will be the same female fitting as in the radiator tank, the other end will be a hose barb. You just tighten that on to the end of the factory cooler lines. The other fitting will screw into the radiator tank, and have a hose barb sticking out to connect the hose.
Plumbing should go like this:
Radiator IN: HOT line from trans into radiator (Undisturbed)
Radiator OUT: Install special hose barb fitting in radiator, run rubber hose to aux trans cooler IN
AUX cooler IN: connect rubber hose from Rad OUT.
AUX cooler OUT: Connect to trans return line using special fitting with hose barb.
FWIW, trannys run line pressures between 90-300psi. Any opportunity to leak or slip apart and they will.
The fittings will look 'sorta like this:
This one goes in the radiator:
This one goes on the trans cooler return line:
A double flare leaves a soft rounded edge:
A single flare does not: