Lead acid batteries (12 volt nominal) are considered charged at 13.2 volt, to charge them over 13.2 volt is required, by time you reach 14.8 volt it can damage the battery, but the higher the volts the faster it charges, as they reach 80% charged then they slow down in their ability to accept charge, so there are three systems used to regulate battery charging.
1) Constant voltage normally around 13.8 volt.
2) Stage charging starts at max output of charger (stage 1) then has a voltage limit 14.4 with VRLA and 14.8 with batteries you can top up, but at the same time it monitors current, at current drops to between 1/10 and 1/20 of amp-hour rating it drops the volts to between 13.4 and 13.8 volts, so the last 10 to 20% of the charging is done at a slower rate. Because it measures current the charger has a range of AH batteries it can charge, too small of a battery and too large of a battery can be over charged.
3) Pulse charger these send a pulse of charge and measure voltage decay between pulses, these are more suitable for when the battery is being charged and used at the same time. Solar panel and wind charger units are normally pulse chargers, also the units used with boats to combine the outputs of two generators and charge multi-battery banks. The CTEK charger I think uses this system, I had a cheap Lidi charger that did this, but wrecked it by putting another charger on the battery at the same time.
So if you have a 13.8 volt power supply that will do the job, even if over 13.8 volt you can use a 7812 voltage regulator and a red LED between Com and real negative. CB power supplies were made using that idea, I use a ham radio power supply and it works A1 when I have the time. What you need to work out is how fast do you want to charge the battery?
The small VRLA (valve regulated lead acid) the valve is simply a rubber cap, when they have been over charged I have been able to add a few drops of distilled water, but really you should not open them. My stair lift has two 12 volt VRLA and there is 29 volts on the charge rail so 14.5 volt per battery, they last around 2 years, I am sure if the volts were dropped the batteries would last longer. It seems therefore that they will stand some abuse, I consider 14.5 volt abuse, I am sure the idea is the lift is used every day so it needs that little extra, but my lift is used once a week to transport the washing.