So please answer to my question: Does this mean that a car charger 12V/6A (i suppose my PSU in the time depth will do the same thing) always adjust the voltage/current (high voltage - lower current or lower voltage - higher current) according to the charging capacity of the battery?
A lead acid battery will draw whatever current its chemical composition allows it to, so as long as the source voltage is higher than the battery voltage, it will draw current, otherwise try to supply current. The 12V tag is obviously just the nominal battery voltage (under nominal load).
I mean, does this mean that my PSU when my battery will get enough charge, will gradually drop down the current demand and raise the voltage to the initial settings ~14.2 ?
Yes, which is why the initial setting should not exceed the voltage a fully charged cell can maintain safely - unless you are able to monitor constantly or use a charger with a builtin detection to shut supply off when charging needs to end.
Any cell -- applying extra voltage will force electrolysis of water as an alternative reaction, worsening charge efficiency and generating gas. Worse still, the gas bubbles block electrolyte, raising resistance and allowing the voltage to rise even further (if otherwise unlimited). This sets the maximum charge rate on a typical lead acid cell.
I know cars that are designed to have up to 15.5V onboard system voltage, but they utilize charge counting to the battery, therefore apply this only with an empty battery and call such a system microhybrid (situational use of the generator). I haven´t heard of exceptionally high battery failures with them, seems to work fine. Besides that, there are always failures rooted in a suboptimal driving profile.
The bench PSU approach has no builtin limitation for a charging process other than the voltage and current limit settings, which is good enough for lead acid and NiMH/NiCd, but not for Lithium cells.
NiMH and NiCd are the ones with weird charging curves (voltage goes up, then curls over slightly as temp rises).
Lithium depends on type, but ordinary LiPo/ion types are CC/CV to cutoff, AFAIK, very easy to use.
Might be wrong on the NiMH and NiCD, haven´t messed around with those...
I thought that the LiPo/LiIon types have no easily detectable deltaV charging end and require thermal monitoring throughout, therefore need charge counting to really utilitze the capacity. I am sure one can get away with keeping it in between 20-80% level, i heard this is done to increase the lifetime/amount of cycles as well.
This is why one really needs a seperate charge logic for every type of battery, it could end catastrophically.