Indeed, there could be multiple issues on your pair of wires, but for info purposes .... a 'Battery contact' fault means your circuit is contact with another circuit, so unless you have more than one circuit in your premises, the battery contact fault can't be in the overhead span from the pole to your house.
Like I say though, that doesn't mean you have two, three, etc different faults on your pair of wires. The older wiring (twin-flex as you refer to it) that is DW3 or DW6, is notorious for having its insulation cracking and allowing an ingress of water, causing a high resistance (HR) fault. Plus, the newer DW (Dropwire) from the pole terminates with the older DW on your house and if the correct crimps haven't been used, this too can create a HR effect.
The thing with the CP's remote test, it only gives the one result ie: loop, dis, battery contact, earth contact, receiver off-hook, unknown termination, faulty line card ..... a battery contact will always be reported above a high resistance fault. Only if it was a full on disconnection (dis) would it be the returned test result.