No, the intelligent (budget-concious) solution in this case is to use the cat3 conductors and any of various schemes to run ethernet over them; starting with the base case that 10Mbit ethernet runs happily on Cat3 wire, and many Cat3 wires are actually fine for 100 Mbit. Those are "free" solutions.
You could put trunking switches on either end and run a bunch of 10 (or 100 if you get lucky) mBit connections in parallel (there will be a lot of pairs - likely 25, so you can run up to 12 wait, you say 50 or 100, so 25-50 - only takes two pair each) for pretty cheap. Except you went and yanked on and broke the wires already.
You can get two-wire "ethernet extenders" that will happily run 40 mBit or more on a single pair. They cost a bit more, usually. Your wire abuse also lowers the odds on those, unfortunately.
Thus, the correct solution for you in this case is to ignore the wire you have been abusing and put a couple of 5 GHz Point-to-Point radios on the buildings where they have a clear line of sight between them.
Or dig a trench and install fiber optic in conduit. That certainly is the "price no object professional approach" and has the benefit of being unaffected by thunderstorms which can wreak havoc with copper inter-building connections. Digging up the old cable is not normally a part of this approach, since it just adds a headache. Calling DigSafe several days before starting is a part of it, though...
When you pull on the wires, you are probably pulling against the telco's strain relief when you snap them. Virtually all "old phone cables" were installed by Ma Bell and she didn't just dump a cable into the ground. Which is to say, yanking the wires out is not going to work.
If the cable is concealed entirely within the wall, then non-metallic sheathed cable (NM) is fine -- the inside of a wall cavity is dry unless you have some serious problems with your house! You only need underground feeder cable (UF) if the cable's buried, put through an outdoor conduit that can get water inside it, baked in the sun, or otherwise exposed to the elements.
Best Answer
Absolutely not. Underground rating presumes physical protection by the surrounding earth.
Undersea cable is designed to be laid on the sea floor. You may have other threats like water-moved rocks, or the footfalls of people or animals that even undersea cable is not made to deal with. There might be no cable rated for that duty.
If the cable is buried so it is both underwater and underground (in reasonable soil near the cable, i.e. not sharp rocks; but then rocks or something above to stop stream bed erosion) -- that would be fine.