Forum adverts like this one are shown to any user who is not logged in. Join us by filling out a tiny 3 field form and you will get your own, free, dakka user account which gives a good range of benefits to you: No adverts like this in the forums anymore. Times and dates in your local timezone. Full tracking of what you have read so you can skip to your first unread post, easily see what has changed since you last logged in, and easily see what is new at a glance. Email notifications for threads you want to watch closely. Being a part of the oldest wargaming community on the net. If you are already a member then feel free to login now.

Let's say I have all Alpha Legion keyworded units in one Battalion detachment. And my second detachment is all Emp's Children.

Can I have the Alpha Legion units use the Legion Trait "Hidden in Plain Sight" while the second detachment can use "Flawless Perfection"?

Yes, when running multiple detachments, each detachment can have a different <Legion> Keyword.

And each detachment can benefit from the rules triggered by having the same <legion> keyword on all the models.

Additional example of what I ran: World Eaters in a Battalion detachment, focused on Berzerkers as troops and a Patrol detachment focused on Noise Marines as troops.

Correct, the reasoning is thus:
If your army is Battle-forged, all INFANTRY , BIKERS and HELBRUTE units in Chaos
Space Marine Detachments gain a Legion Trait, so long as every unit in that Detachment
is from the same Legion
- Legion Traits

All Units in All Detachments, ensuring different Traits are separated into different detachments.