Which among the circuit is the most efficient for battery backed up systems?
Efficient depends on battery life and reliabilty.

Version 1: charges the battery and the battery gives the power to the circuit. Zero transition time in case main power fails, cause load is always served from the battery. High strain on battery.

Version 2: charges the battery but also gives power from main line to load. Battery also charges, but only when it leaks or was drawn on main power fail. Normal strain on battery. Fast take over in case of main power fail.

Version 3: Battery only charged if not full. Main power feeds load directly. Bit slower take over, due to relay needs to decharge (If its a normal coil based relay). Close to no strain on battery.

At least for starters this should be adequate.