What I was wondering is can the off-hook action of a phone be used to trigger a call, i.e.
lift handset at site "A" - Obi recognises this and dials the number of the Obi at site "B" and vice versa.
Yes, it's called Hot or Warm dialing. The OBi will automatically dial a particular number immediately (Hot) or after some period of time (Warm) after the handset goes off-hook.
Hot:
Physical Interfaces -> PHONE Port -> DigitMap : (<S0:**9200123456>|...)
Warm:
Physical Interfaces -> PHONE Port -> DigitMap : (<S4:**9200123456>|...)