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What I mean is are they part of the Black Legion Faction and therefore must take the VotLW upgrade? For example for the Raptor detachment do I need to take VotLW if theyre just going to be part of a regular CSM force?

Its not black legion at all

TH is plain old csm, which allows both BL and CS to use the formations/detachments as their own too. Only thing the supplements can't affect is the roll on the warlord trait. When using the TH decurion you only get to re-roll on the vanilla table. It doesn't stop you from rolling on the supplement tables though.

If they were BL formations they would say so under the restrictions section of the formations.

They don't however so you're free to use them with any supplement or the regular codex.

This detachment does seem like it's meant to be paired with the Black Legion, as VotLW becomes free which was a major requirement in Black Legion Detachments. However it also erases one of the benefits, namely Chosen as Troops (technically this still exists but doesn't do anything since there are no troop choices). One benefit is that you get to use the BL Relics and Warlord Traits with this without the VotLW tax.

Neither supplement gains anything at all from being used with the decurion or formations. CS keep their fear and possessed table but can't benefit from the free votlw while black legion becomes completely pointless since having chosen as troops does nothing and whether you play vanilla with VotLW or BL with VotLW makes no difference at all. All you get from using supplements is access to one set of relics or another. Well minus the favoured where CS get screwed over.
Black Legion would gain a lot if you could benefit from the free VotLW with their own formations. Unfortunately that doesn't work. You're stuck with the decurion if you want free VotLW and any formations you add are technically separate.