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I found it in the Contacts app but not if you access their profile from a convo in Messages.

Doesn't show up for me either when going in through the message app. The only way is through the contacts app like dcp10 mentioned.
 
I found it in the Contacts app but not if you access their profile from a convo in Messages.
It was like that in iOS 8 I believe.

Same if you find the contact through the Phone App, you can edit but deleting is only done through the Contact App.
 
If it's not in iOS 8 like the last couple were able to confirm, that tells me it's intended to only be deleted from contacts app.
 
You're right. Oddly it's only from Favourites & Recent tab that you won't be able to delete a contact (in the Phone App) in the Contact tab you will be.. Uh!

I just went through a good portion of my contacts and it seems contacts that were added from Facebook don't show the red "delete contact" at the bottom when you tap edit. All other contacts will.
 
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