They both can use an absorb phase but to be healthy a Lead Acid battery must be kept at the Absorb phase longer. Yes Lead Acid is less efficient because you have to put in more than you took out. A Lithium battery does not require an Absorb phase and could be terminated at any voltage and be left there with no damage. A lead acid battery has to see a prolonged absorb phase to stay healthy. That extra time does not result in much increased capacity hence the less efficient charge characteristic
Any charger is a fixed voltage charger as soon as you set the constant voltage setting. However what you set it to is not necessarily what it puts out as soon as you connect it to a battery. For example if my target voltage is 13.6 volts but the voltage of the battery is 13 volts, as soon as you connect them the battery charger or power supply will try to pump as much current as it can to get the voltage to 13.6 but the charger or power supply has a current limit which is why it is called the constant current stage. As the battery is charged, the voltage at the charger/battery increases until finally the constant voltage setting is reached.
Some chargers can be set for an absorb time of zero so in that case that setting would be a single stage of constant current until the voltage setting is reached. Other settings for absorb (Constant Voltage} can be to terminate Absorb after so many minutes or until a minimum number Amps is reached. A power supply typically will not have those choices so it will keep going at the fixed voltage until the current drops to zero whether it is attached to a Lead Acid battery or Lithium battery.