RickyJ,
You are correct with your assumptions of the handover of aircraft between Ground & Tower controllers. But there are a few wrinkles around the world.
1. If an aircraft taxiing with Ground has to cross the active runway it should switch to Tower to request clearance to cross, then report runway vacated and switch back to Ground.
2. At some airports this frequency switch is negated by the aircraft staying on Ground, requesting clearance to cross, then Ground coordinates with Tower, approves the crossing, aircraft reports vacated, Ground informs Tower.
3. At a smaller number of airports permission is also required to cross inactive runways.
It has been a long time since I went to KSAN so I don't know if they use option 1 or option 2 for north side traffic.