I want to know if anyone else experience the problem I described: camera thinks that Eneloop XX batteries are low after only a few shots.
From what I've observed, the batteries are not low. Proof of that is that I can continue shooting for hundreds of shots beyond the indicated low battery point. How many hundreds? I don't know for sure, since I've never shot until the batteries are completely dead.
Same behavior with video. Low battery indicator comes on after 30 min or so (on and off, not continuous recording), but I can continue recording for at least another hour or two.
This is normal, simply how it works, and is not a problem. You have no battery meter for NiMH. It is not just related to LSD Eneloops.
NiMH nominal voltage is lower (like 1.3 volts) than Alkaline (1.5 volt) or Lithium AA (1.6 volt). That is a misleading statement, because the NiMH voltage holds a more constant voltage over their usage lifetime, where the other types start falling fast, and much of their lifetime have lower voltage than the NiMH.
The lower voltage is not an issue in flashes, which have internal power converters to convert the low battery voltage to be over 300 volts flash capacitor voltage, which drives the flash tube. The flash recycle time is charging that capacitor again. The 300+ volts is regulated, not allowed to go higher. High current capability is all important to the flash battery (NiMH are ideal). Cameras are high current too (compared to a mp3 player), but very much less important than to the flash recycle.
The battery meters in electronic gear are just simple voltmeters. An alkaline starts at 1.5 V, which is full charge. Towards the end, it is 1.0 V, and it's dead. Around 1.25 V is half charge. The meters are voltmeters, and are marked that way. Alkalines can go down to 0.9 volts, but most electronic circuits quit at 1.1 volts.
But NiMH start out at 1.3V, and these dumb simple meters pronounce them half charge when fresh. Of course, they continue to just keep on working, because that is their full charge state. The meter just does not know about them. And the NiMH constant voltage (which is a plus) is not going to let the voltmeter see a change anyway. They "appear" to stay at "half" state the entire time, until they suddenly go dead. So a voltmeter is not meaningful to determine NiMH charge. I am not aware of any way to determine NiMH state of charge (until you either test by discharging it, or you test to see what it takes to recharge it).
Lithium Ion camera batteries are 3.7V and rechargeable, a very different chemistry than Lithium AA (1.6V and not). But due to their constant voltage advantage, camera and laptop Lithium Ion batteries have chips in them (coulomb counters), primarily for lithium safety, and to meter the usage and report a meaningful charge state. But NiMH do not.
The NiMH can deliver much more current than either alkaline or lithium, so NiMH is ideal for a flash.
The Lithium AA cells were previously called Advanced, and next version is now called Ultimate (Energizer). The chemistry might be able to deliver high current, but which was a safety problem (lithium fires). So Advanced batteries (AA) were artificially limited to peaks of 2 amps for safety reasons (fire). This meant they had a slow recycle time in flash units. Later they increased this limit to 5 amps (called Ultimates), and now they recycle flash as fast as Alkalines (but NiMH are still noticeably faster).
Four Flash Photography Basics we must know - Powering the flash - Batteries