Fall damage death messages don't show block player fall from correctly (ladders, vines, water)

The bug

Death messages, such as:

• <player> fell off a ladder
• <player> fell off some vines
• <player> fell out of the water

don't show instead it is just <player> fell from a high place

How to reproduce

1. Make a very tall tower on which one side has a ladder and the other side has vines
2. Climb to the top of either side, and fall off, killing yourself
3. Death message <player> fell from a high place displays

See also this reddit post: http://www.reddit.com/r/Minecraft/comments/2wpmer/help_me_die/

Code analysis

Code analysis by [Mod] NeunEinser can be found in this comment