Reported: Some quests, such as the [Lv6_Order] quests, show no objective in the Aim section and no progress message on a kill, yet the kills still count and the quest can be completed.
Found: The server does track these kills and does send a progress update for each one, which is why the quest completes. What it gets wrong is the quest's starting state. When a quest is handed out, the server tells the client which of the quest's three objective slots are in use. It takes that from a short built-in list covering the first few class quests, and every quest not on that list, the level-up quests included, is announced with all three slots in use. The [Lv6_Order] quests define one objective, so the client is told about two objectives that do not exist. That the client then shows none of them is our best explanation, not yet confirmed against the client.
Planned fix: Announce each quest with exactly the objectives its own quest data defines, instead of the built-in list. Work on it is starting.
To verify: Once the fix is out, reach level 6 on a new character and open the new [Lv6_Order] quest: the Aim section should list its objective, and each kill should show a progress message.