When creating a new damage type it is not enough to /reload, like it is for other custom data like loot tables, tags, etc, but a world reload is needed to see changes to the custom damage type.

Reproduction:

Create a datapack with a new damage type

{ "exhaustion": 0.1, "message_id": "arrow", "scaling": "when_caused_by_living_non_player" }

Use the command /reload to reload the datapack

The new damage type will only appear once the world is reopened

Then change the field "message_id" from "arrow" to "cactus", this should make it so that the death message is the cactus death message instead of the arrow.

Use the command /reload to reload the datapack

Use the command /damage to deal lethal damage with your newly created damage type, it is still the arrow death message. 

If you reopen the world and try dealing lethal damage with the new damage type, the death message should have changed to the cactus one.