Not much to chose between them. The more complicated one reduces the charge to a trickle which is only important if you want the charger to be connected 24/7.
Both will handle a range of batteries up to about 15V, but both are unnecessary if you just want to do occasional charging.
Lead acid batteries can be charged from a constant voltage and will adjust the current by themselves.
If you have a variable dc power supply, just set it to about 1.2 x the nominal battery voltage and connect up (eg 7V for 6V battery, 14V for 12V battery).
You won't be able to hurt a car battery, but sealed lead-acid batteries have a maximum charge rate so set the current limit on the power supply to this.