the relay or the immobiliser
Well, on a B110, there is no starter relay except the solenoid on the starter. The solenoid has the relay built-in (the copper bits discussed above).
Just take the starter apart and look at it. If the solenoid contacts are bad, the starter might work fine w/o a load on it. But when trying to turn the engine, the contacts spark out. Just like a dirty battery terminal, they might run the headlights, but stop altogether when you crank the engine.
Not sure what an immobiliser is, what's another word for that?
He seems to think the voltage coming to the starter might be low
You can check like this: put your voltmeter directly on the battery terminals. You should get near 12 volts. Now put the voltmeter earth lead on bare metal surface of engine (like the aluminum manifold) and the other lead on the big solenoid terminal opposite the battery cable. It should read zero. Have someone attempt to start it (be careful of moving parts). The voltage should go from zero to 9-12 (?) volts or thereabouts. If it stays at zero or close to zero with the key held at start, that would indicate a bad connection somewhere. In start, the solenoid coppers should be directly connecting the battery cable to the output cable, and it should read battery voltage. Most likely the bad connection is inside the solenoid, but could also be a bad earth or bad cable connection.