Specifically this only happens when using the {self} selector as part of the command. When a command without {self} is used, the command triggers properly from all damage sources.
Commands with a {self} selector do still trigger correctly if the mob takes damage from the environment, such as fall damage, fire, etc.

Examples:

• Will trigger from any damage, including player-dealt damage (working correctly):

{
	"name": "Ouch1",
	"type": "PASSIVE",
	"subtype": "OFFENSIVE",
	"activationType": "onDamage",
	"command": "me says Ouch!"
}

• Will trigger from environmental damage, but NOT player-dealt damage (not working correctly):

{
	"name": "Ouch2",
	"type": "PASSIVE",
	"subtype": "OFFENSIVE",
	"activationType": "onDamage",
	"command": "execute as {self} run me says Ouch!"
}

This isn't an issue with only execute as {self}, using effect give {self} ... results in the same behavior, whereas when I use effect give My_Username ... it works correctly. It seems to be an issue with using selectors in the command.

Fixed in latest release