I'm particular to a charger (trickle or otherwise) that puts out a steady 13.25 to 13.75 at all times. That being said... I'm not going to toss a "battery tender" type charger if it only charges at 12.5 or so. I figure in the long run, ANY amount of charge going into a battery is better than it sitting and slowly discharging.
I would consider 12.5 damagingly too low. Don't forget that this is a charger that is not diode isolated from the battery. So if the battery is fully charged (Rest voltage 12.8 to 12.9 volts) and the charger is at 12.5, then the charger will actually DISCHARGE the battery.
I would say the bare minimum fairly safe voltage is about 12.8 or 12.9
I would much rather see 13 to 13.2 for long term float charging, and higher for other charge modes.
If the charger is diode isolated, then sure, even 12.5 volts won't really hurt, but it probably won't help much either unless the battery is already so low that its being damaged. In that case the 12.5 volt charger won't prevent damage, it will only keep the damage to moderate damage instead of severe damage ... whoo hoo.
All that is kind of moot though, since it appears that these chargers are about 12.8 or above...
I'm slightly embarrassed to admit it, but while in high school, I worked in the garage at the local Montgomery Wards (not the highest of recommandations for quality wrenching) and I was "trained" to become the electrical systems specialist (WhooHoo!). One of the things I specifically remember is that lead-acid batteries do develop "charge memory". If the battery is constantly kept at a steady charged state without discharging for long periods of time, it will develop a memory for taking a light charge and no more. It will reach a point that even when it is fully discharged, it won't take a good charge afterwards. It is much better to discharge it occasionally, and recharge again afterwards. The cycling up and down keeps the lead plates healthy and capable of current output for the longest period of time. (full battery life).
The sulphuric acid (electrolyte) floods the lead plates and the discharge/charge process causes the sulphate ions to transfer into and out of the lead plates depending on the current flow. Sitting in either a fully charged or fully discharged state for long periods of time tends to permanently "clog" the pores of the lead plates, drastically reducing the transfer of the sulphate ions.
Regarding float charging the battery being bad: I have NEVER heard of anything like that. Everything I have ever read about lead acid batteries from both battery manufacturers and charger manufacturers says to store by either float charging, or leaving unconnected and periodically topping up.
The details of the required voltages and how often to top up vary somewhat depending on many factors, but the overall general recommendations are ALWAYS the same.
The only thing I can imagine is that the person who taught you was really mangling the idea of something called "electrolyte stratification". This can happen for both batteries being stored open circuit, and for batteries being float charged. Although it can happen to automobile and motorcycle batteries, it is mainly a problem for physically very large batteries (like on WWII submarines, or off grid power systems)
If a battery sits for a long time, the electrolyte will form regions of different specific gravity at different heights in the battery. Just like a layer of oil will float on top of a layer of water, some of the lighter electrolyte seperates out, and will float on top of the heavier electrolyte. One of the two regions (I forget which) corresponds to discharged electrolyte. Where that electrolyte contacts the plates the plates will sulfate, just as if the battery was discharged, even though the battery as a whole is still fully charged.
The cure is to periodically charge the battery at a high rate, and make it gas (bubble). The gassing mixes the electrolyte thoroughly, so that regions of different density don't get a chance to form and then seperate. But discharging is NOT required to prevent it. And for batteries that are not specifically made for deep cycling, any significant discharge WILL reduce the life of the battery.