As usual, Doctor Who always plays fast and loose with the rules, but there is a bit of consistency here over how this regeneration works.
The general idea appears to be, if you can kill a Time Lord while they are regenerating, then their body won't be able to recover. It appears that the body needs to have some amount of life in it for the regeneration process to kick in, and it can be short circuited. Besides Lake Silencio, we've seen other examples of this:
- The Eighth Doctor "died" in a crash and would not have regenerated without the Sisterhood of Karn intervening.
- The Seventh Doctor's regeneration was delayed because he was put in cryostorage, slowing his bodily functions
- Missy has claimed that 8 snipers using conventional Earth weapons could kill her (though she could be lying)
- The Eleventh Doctor was dying from Judas Tree poison, which blocked his regeneration, until River intervened.
- Missy died because The Master "overloaded" her body worse than her regeneration energy could repair it.
On the other hand, we have also seen how a Time Lord mid-regeneration has remarkable healing powers: the Tenth Doctor regrew and entire hand that had been lopped off while he was still regenerating from being the Ninth Doctor.
If we follow that pattern, than the difference between the Eleventh Doctor and River is that the gunshots River took didn't kill her, so her body was able to repair itself. The weapon River used on The Doctor as Lake Silencio, on the other hand, appeared to kill him very quickly, giving his body no time to heal between "deaths" and thus, killing him permanently.
Of course, we now know that the whole scene was staged. For example, we know that The Doctor would not have regenerated had he really been shot, since he had used his regenerations all up by that point. So we still don't have a clear picture of what happens when you kill a Time Lord mid-regeneration.
It's possible that River was just wrong, or just guessing about how he died. River seems fairly knowledgeable about Time Lords, but she has also been known to lie.